arXiv:astro-ph/0307154v1 8 Jul 2003 eta est up.Ee h eyhgetresolution highest very the Even ( cusps. density central l 20,hratrSV)ageta oteitn ro- ( existing NFW-like most with consistent that et al. also argue Swaters et are SMVB) and Bosch curves hereafter (2001), tation den (2002, van Swaters hand, & al. other & Bosch den the Blok, van 2002; On de (2000), Bosma Weldrake, 2003). & 2002; de Blok Borriello Walter 2001; de & Salucci 2001; Walter, & Rubin Salucci, Borriello & 2001; McGaugh, al. obser- Blok, et the by Blok claim allowed (de authors are vations Many cores constant-density profiles. only density that ob- matter in the of dark consensus shape served of actual the lack on 1999b). community disturbing observational al. the a Burkert et remains 1994; Moore there 1996; Primack However, White & & Frenk, Flores the Navarro, over (e.g., 1995; been observers years and has several theorists simulations past nu- both (CDM) by by discussed predicted Matter widely profiles Dark density Cold the merical and brightness low-surface galaxies and (LSB) dwarf of profiles density matter eut li-ulet,Arm ainn(01 found (2001) Carignan ρ & Amram, Blais-Ouellette, result; h tla ikcnrbtost h oaincre) Bo- curves), rotation the to contributions disk stellar the 1 nin nvriy aeUiest,adteNtoa Opti National the and 2 University, Yale University, Indiana rpittpstuigL using typeset Preprint in publication for Accepted nvriisfrRsac nAtooy n.(UA ne c under (AURA) Inc. Astronomy, in Research for Universities . ae nosrain are u tteWY bevtr.T Observatory. WIYN the at out carried observations on Based iiigAtooe,Kt ekNtoa bevtr,Nati Observatory, National Peak Kitt Astronomer, Visiting ∝ h paetdsgemn ewe h bevddark observed the between disagreement apparent The 0 c tde ontse ob ovrigo single a on converging be to seem not do studies pc) 100 r − 0 . 3 ujc headings: Subject Ovlct ed fNC27 ge ihnterucranis ihat a with uncertainties, their s within km agree 5.3 2976 of NGC velocities of fields velocity CO h hp ftedniypol ftedr atrhl ihgo pr good with a halo follows matter 2976 NGC dark of the distribution of mass near-infr profile matter) and dark density optical plus the (baryonic multicolor of our shape with the along observations, these 0 vrteetr bevdrgo.Amxmlds tyed nuprlimit upper an yields near fit a disk with maximal consistent ( A shallower, ratio region. light pr even observed motions is entire radial halo observed the the over dark that the assuming to kpc, attributed 1.8 of radius a to M O h ihsail( spatial high The CO. ailmtosta r slrea 0%o h oainlvlcte a velocities rotational the of % 90 as large as two-dimensiona are than errors a that systematic studies, motions matt to curve radial dark vulnerable rotation cuspy more hamper a far can contain are that not effects does 2976 systematic NGC the profile, density the of ρ . DM nNC30 and 3109 NGC in 10 ∗ ehv bandtodmninlvlct ed ftedafspiral dwarf the of fields velocity two-dimensional obtained have We /L M ∝ IHRSLTO ESRMNSO H AKMTE AOO NGC OF HALO MATTER DARK THE OF MEASUREMENTS HIGH-RESOLUTION K smnatobree.d,[email protected] [email protected], ⊙ r /L − > 0 A . ⊙ 0 1. 01 T M K . E 19 hrfr,idpneto n supin bu h tla dis stellar the about assumptions any of independent Therefore, . tl mltajv 11/12/01 v. emulateapj style X ∗ introduction . /L ohaD Simon D. Joshua M h srpyia Journal Astrophysical The M aais ieaisaddnmc aais spiral galaxies: — dynamics and kinematics galaxies: ⊙ K akmte aais wr aais niiul(G 52 G 2 NGC 2552; (NGC individual galaxies: — dwarf galaxies: — matter dark /L ∗ ρ f0 of ) /L − ∝ ⊙ 1 K K r VDNEFRASALWDNIYPROFILE DENSITY SHALLOW A FOR EVIDENCE ∼ taypsto ntegalaxy. the in position any at − . eateto srnm,Uiest fClfri tBerke at California of University Astronomy, of Department ≤ 09 h akmte est nrae ihrdu,wihi nhscl A unphysical. is which radius, with increases density matter dark the 0 . 5p)adseta 1 ms km (13 spectral and pc) 75 5 − +0 0 cetdfrpbiainin publication for Accepted . 0 nI 54(ignoring 2574 IC in 19 . . 15 08 M 2 M let .Bolatto D. Alberto , ⊙ ⊙ /L /L ⊙ ⊙ 0 apelHl,C 94720 CA Hall, Campbell 601 K K ρ a srnm Observatory. Astronomy cal h akmte est rfiele between lies profile density matter dark the , ∝ icuigsseai netite) ihtecva htfor that caveat the with uncertainties), systematic (including oeaieareetwt h ainlSineFoundation Science National the with agreement ooperative r nlOtclAtooyOsraoy hc soeae yth by operated is which Observatory, Astronomy Optical onal ABSTRACT − eWY bevtr sajitfclt fteUiest fW of University the of facility joint a is Observatory WIYN he 1 ) 1 ,[email protected] [email protected] [email protected], u, h srpyia Journal Astrophysical The ∼ hwcnrldniypolsta r ossetwt any with consistent are that between profiles shape density central show isi h aatesle.Frteprmtr ftypical of parameters ambigui- the ( For reflects observations galaxy themselves. observers dwarf/LSB data among the in consensus ties of lack the constant-density essentially density an a contains has halo. 6822 4605 NGC NGC that that showed (2002) profile al. et latto infrH for tion nua eouin( resolution angular lto and olution ( blt otesseai rbesta a ffc single a H affect example, can For vulner- that our problems tracer. reduces systematic wavelengths the different to ability three or field Ob- velocity two the at targets. of tracers as independent galaxies completely dwarf serving nearby 5) and photometry, ainlcalne.Orpormicue )two- 1) includes obser- (H program the optical at overcome Our obtained to fields velocity techniques dimensional of challenges. number vational a bines iee C) n etmtr(H centimeter and (CO), limeter . 2 dmLeroy Adam , h eetsuyb MBsosta,i ag part, large in that, shows SMVB by study recent The eadesti rbe ihanwsuyta com- that study new a with problem this address We ms km 2 − 0k s km 10 1 n ms km 2 and ρ ∝ − dso ht1 ogltrtto curves rotation longslit 1) that show nd 1 eoiyfils )NC27 contains 2976 NGC 2) fields, velocity l I r vd ospot h est profile density The support. no ovide − ∼ eoiyrslto and resolution velocity nefrmty,te n that find they interferometry), − rhl.W loivsiaesm of some investigate also We halo. er 0 rdiaig lo st measure to us allow imaging, ared 1 ycntn est fdr matter dark of density constant ly r . 1 ,4 utclrotcladnear-infrared and optical multicolor 4) ), 65 0 ρ ml ai,ad3 h H the 3) and radii, small t ′′ 2 TOT n edaee l 20)determined (2003) al. et Weldrake and , cso.W n httetotal the that find We ecision. n e Blitz Leo and , and eigfrlnsi H longslit for seeing − pclsatrbtentetwo the between scatter ypical oteKbn tla mass-to- stellar K-band the to aayNC27 nH in 2976 NGC galaxy 1 ∼ epciey eouinof resolution respectively) , ∝ ley r − r α 5 1 1 − rtefntoa form functional the or k ′′ . 0 eoiyfilscnb distorted be can fields velocity ,3 ihseta resolution spectral high 3) ), . 27 ± 0 . ρ 09 ∼ DM I aeegh,2 high 2) wavelengths, ) oe a out law power 0k s km 50 ∼ ∝ α 7)— 976) 15 r 2976: − bevtos and observations, . ′′ ssuming 0 u . isconsin-Madison, − 17 nua resolu- angular α α soito of Association e 1 otgalaxies most and and and eoiyres- velocity α ,mil- ), 2 Simon et al. by extinction, or by large-scale flows that are associated combined with archival 2MASS near-infrared images, en- with star formation, while existing H I data generally suf- ables us to accurately model the stellar disk. fer from beam smearing. Two-dimensional velocity fields In the following section, we describe NGC 2976 and our also represent a major improvement over the traditional observations and data reduction. In §3, we model the stel- longslit spectra, making the effect of positioning errors lar and gaseous disks. In §4, we derive the rotation curve negligible and allowing us to account for simple noncircu- of the galaxy and the density profile of its dark matter lar motions. High angular resolution is important because halo. The analysis routines that we use are presented in the central cores described in the literature have typical more detail in Appendix A. We discuss our results and radii of ∼ 1 kpc, which corresponds to an angular size of their implications in §5. In §6, we describe some system- 20.6(d/10Mpc)−1 arcseconds. In order to resolve this size atic uncertainties that can affect rotation curve studies, scale and minimize the impact of beam smearing on our and test the robustness of our results against them. We conclusions, an angular resolution element several times present our conclusions in §7. smaller is required. High spectral resolution is also bene- ficial because it results in more accurate rotation curves. 2. target, observations, and data reduction Finally, our multicolor photometry plays a crucial role in allowing us to attempt to realistically model the rotational 2.1. Properties of NGC 2976 contribution from stellar disks instead of simply guessing NGC 2976 is a regular Sc dwarf galaxy located in the an appropriate mass-to-light ratio and assuming an expo- M 81 group. Karachentsev et al. (2002) measured a dis- nential disk. tance of 3.56 ± 0.38 Mpc using the Tip of the Red Giant Target selection also has important effects on the Branch (TRGB) method, and the Tully-Fisher distance is strength of the conclusions we will be able to draw. We fo- 3.33 ± 0.50 Mpc (M. Pierce, private communication). We cus on very nearby objects (D < 10 Mpc) in order to max- adopt a distance of 3.45 Mpc, which sets the conversion imize our physical resolution. Dwarf and LSB galaxies are between physical and angular scales to 16.7 pc arcsec−1. the preferred targets for this type of study because they are NGC 2976 has absolute magnitudes of MB = −17.0 and −1 presumed to be the most dark-matter dominated galaxies. MK = −20.2, a heliocentric velocity of −0.8 ± 1.8km s , −1 (Note that in this paper when we refer to dwarf galaxies, an inclination-corrected H I linewidth W20 =165 km s , 9 we mean high-mass dwarf irregulars and low-mass spiral and a total mass of 3.5 × 10 M⊙, so it is somewhat less lu- galaxies, not dwarf spheroidals or ellipticals.) LSB galax- minous and less massive than the Large Magellanic Cloud. ies, though, tend to be relatively distant and are necessar- The low systemic velocity is not a problem for our obser- ily quite faint, so they are difficult to observe with suffi- vations because the galaxy is located at high Galactic lat- cient resolution and sensitivity. Dwarf galaxies, in compar- itude, where there is little Milky Way CO emission, and ison, are both bright and plentiful in the nearby universe. no Galactic Hα emission is visible. In optical and near- Dwarfs are traditionally presumed to be dark-matter dom- infrared images it is clear that NGC 2976 is a bulgeless, inated at all radii (Carignan & Freeman 1988; Carignan unbarred, pure disk system (see Figure 1), which makes it & Beaulieu 1989; Jobin & Carignan 1990; Martimbeau, an ideal galaxy for mass modeling. Carignan, & Roy 1994). However, the observations upon which this assumption is based were made at low angu- 2.2. Hα Observations and Reductions lar resolution. Higher resolution observations of the in- ner regions of dwarf galaxies show, as we discuss later, Our Hα observations were obtained on the nights of that stars can dominate the kinematics of dwarf galaxies 2002 March 20-21 at the 3.5 m WIYN telescope with in their inner kpc (e.g., Broeils 1992; Swaters 1999; Blais- the DensePak instrument. DensePak is an array of 94 ′′ ′′ ′′ Ouellette et al. 1999; Bolatto et al. 2002). Comparable 2. 8-diameter fibers, fixed in a 30 × 45 rectangle with ′′ data for LSB galaxies are scarce (although see Swaters, a fiber-to-fiber spacing of 4 (Barden, Sawyer, & Honey- Madore, & Trewhella 2000; Swaters et al. 2003), but it is cutt 1998). Five of the fibers are broken and four are sky possible that reliance on low-resolution observations (e.g., fibers, placed at fixed positions outside the main array. de Blok & McGaugh 1997) could have caused an over- Thus, there are 85 data fibers covering almost the whole statement of the case for dark matter domination in these instrument footprint (see Figure 2). The fibers feed into galaxies as well. Future studies of LSB galaxies, featuring the Bench Spectrograph, which we used in its echelle mode −1 two-dimensional Hα spectroscopy and/or ≤ 100 pc reso- to yield 13 km s velocity resolution over a 180 A˚ range 2 lution H I observations, if feasible, are desirable both to centered on Hα. The detector was a 2048 SITe T2KC investigate this question and to clarify the severity of the CCD. cusp/core problem. We observed the galaxy at 13 positions to cover most In a previous paper, we reported on a rotation curve of its optical extent (see Figures 3 and 4a). The galaxy study of the dwarf spiral galaxy NGC 4605 (Bolatto et al. was not visible on the guide camera at the telescope, so 2002). In this paper, we present a similar, but improved, we acquired the target by offsetting from a nearby bright study of a second nearby dwarf galaxy, NGC 2976. As star. Each subsequent position on the galaxy was ob- before, we use high-resolution CO interferometry to study served by making a blind offset from the previous posi- the inner velocity field of the galaxy, but we have also ac- tion. Integration times at each position were between 20 quired high-resolution two-dimensional Hα data (instead and 70 minutes, with just a single 20-minute exposure at of longslit observations) to supplement the CO and extend most positions. Consecutive exposures at the same posi- the velocity field out to larger radii. In addition, we have tion were reduced separately and then coadded. Two of obtained multicolor optical imaging of this galaxy, which, the fields were observed on both nights, and one field was observed twice on the same night, but five hours apart. Dark Matter Halo of NGC 2976 3

In these three cases, instead of assuming that the posi- searched for the highest value of the cross-correlation func- tions observed were the same for the later observations tion. We estimate that the accuracy of the positions de- as they were for the earlier ones, we analyzed the frames rived with this method is 1′′. The algorithm failed for one entirely independently. We therefore had 16 observations field, because only 15 of its fibers contained detectable sig- of NGC 2976, yielding a total of 1360 spectra, of which nal, and very little emission was visible at that location in 1087 contained Hα emission at a level of 3σ or higher. the image. For this field we assumed that the offset from Based on comparison with adjacent fibers that contained the expected position was the same as the one we measured brighter emission at similar velocities, we also judged that for the preceding exposure. Since this field is located ∼ 2′ 9 spectra containing emission at a significance level be- from the center of the galaxy, an error of a few arcseconds tween 2.2σ and 3σ represented real signal. The median in its position is unlikely to be important. For the other 15 detection level in the 1096 spectra that contained emis- fields, the algorithm gave a smooth, well-defined peak with sion was 27σ in integrated intensity, or 12σ at the peak of a cross-correlation coefficient between 0.81 and 0.996. We the line. verified the results of the cross-correlation by finding the The DensePak data were reduced in IRAF3, using the location of the minimum rms difference between the photo- HYDRA package. We subtracted a bias frame, removed metric and spectroscopic fluxes. This position was always cosmic rays, interpolated over bad columns, and then ex- within 1′′ of the cross-correlation maximum. Fourteen of tracted the spectra with the task dohydra. The trace and these 15 fields are located within 6′′. 9 of their expected response function for each fiber and the relative transmis- positions, and the other differs by 11′′. 3. sion efficiencies were derived from a set of flat field images, and wavelength calibration was provided by spectra of a 2.3. CO Observations and Reductions CuAr lamp. Night-sky emission line wavelengths from Os- terbrock et al. (1996) and observations of a radial velocity Our 12CO (J =1 → 0) observations were acquired using standard star were used to check the wavelength scale. the B, C, and D configurations of the 10-element BIMA After extraction and wavelength calibration, we averaged array (Welch et al. 1996) between April 2001 and March together the four sky fibers, leaving out any sky spectra 2002. The total integration time was ∼ 80 hours, much of that were contaminated by emission lines from the tar- which was in the most extended (B) configuration. The ′′ get galaxy. We then removed a linear baseline, performed BIMA primary beam has a half-power diameter of ∼ 100 , a Gaussian fit to the averaged sky emission near Hα, and and we found CO emission spanning this entire width, ′′ subtracted the fit from all of the data fibers. Some spectra including a cloud outside the primary beam at r = 70 contained noticeable residuals at the wavelength of the sky (see Figure 1b). For our observations, the spectrometer − Hα line after this subtraction. Sky residuals are easily dis- was configured with 2 km s 1 wide channels and a 260 − tinguishable from real signals because they are unresolved km s 1 bandpass. The individual tracks were calibrated, and always located in the same four pixels. If the residual combined, imaged, and deconvolved using the clean algo- overlapped with and was comparable in strength to the Hα rithm within the MIRIAD package. The tracks were then ′′ ′′ emission from NGC 2976, the spectrum was discarded (29 combined with natural weighting to create a 5. 2 × 6. 0 spectra were thrown out because of this consideration). (87×100 pc) synthesized beam with a position angle (PA) ◦ This only occurred in places where the galaxy velocities of −31 . The rms noise of the individual planes of the − − were about −17 km s−1 (see Figure 4). Individual frames datacube is 24 mJy beam 1 in each 2 km s 1 channel. of the same field were then averaged together (except for An integrated intensity contour map is displayed in Fig- the cases noted above), weighted by exposure time if it was ure 1b, and a first moment map produced from a masked clear, or signal-to-noise ratio if there were clouds during version of the datacube is shown in Figure 5b. Because the exposure. Velocities were calculated for each fiber by the signal in a single channel was relatively weak, we used fitting a Gaussian to the observed Hα emission. Typical the first moments to represent the velocity at each position linewidths are 34 km s−1, and the median uncertainties instead of attempting to fit Gaussians to the line emission. − on the Gaussian fit centroids are 0.77 km s−1; some fits Typical uncertainties in the line velocities are 3 km s 1, − have uncertainties as small as 0.04 km s−1, and a few are and typical linewidths are 10-15 km s 1 across most of the − as large as 23 km s−1. galaxy, although some lines are as wide as 35 km s 1 near It was obvious from comparing frames that were taken the center. several hours apart or on different nights that the telescope positioning accuracy for our observing procedure was only 2.4. Optical and Near-IR Imaging and Reductions ≈ 5′′. We therefore designed an algorithm to determine the absolute positions that were observed based on our Hα We observed NGC 2976 with B, V, R, and I filters at image of NGC 2976, which is displayed in Figure 3. We the 1.8 m Perkins Telescope at Lowell Observatory on the sampled the Hα image with simulated “fibers” of the same photometric night of 2002 February 11. The detector was a 20482 Loral CCD with 15 µm pixels and a 3.′2 field of size and location as the DensePak fibers, and added up ′′ the flux in each simulated fiber. By cross-correlating this view, and the seeing was ≈ 1. 4. We used exposure times set of photometric fluxes with the observed spectroscopic of 600 s in B and 300 s in V, R, and I and observed three fluxes (integrated over the Hα line), we could measure the overlapping positions to cover the full extent of the galaxy. similarity between them. We repeated this process at a A three-color composite of these images is displayed in Fig- grid of positions around the expected pointing center and ure 1a. To extend our set of images to the near-infrared, we used the 2MASS JHKs Atlas images of NGC 2976. The 3 IRAF is distributed by the National Optical Astronomy Observatories, which is operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under cooperative agreement with the National Science Foundation. 4 Simon et al.

2MASS images are 8.′5×17′ and have 1′′ pixels, adequately value of 143◦, identical to the cataloged PA of the galaxy sampling the ≈ 3′′ seeing. (de Vaucouleurs et al. 1991, hereafter RC3). The ellip- The optical data reduction, done in IDL, consisted of the ticity ǫ, which is related to the inclination angle via the 2 2 2 2 following steps: overscan subtraction, dark subtraction, formula cos i = [(1 − ǫ) − (1 − ǫmax) ]/[1 − (1 − ǫmax) ], flatfielding, and cosmic ray removal. Several bad columns where ǫ ≡ 1 − b/a, a and b are the major and minor axis were fixed by adding or subtracting a constant so that their lengths, and ǫmax =0.8, varies from ∼ 0.4 to ∼ 0.7 in the median values matched those of the surrounding columns; inner part of the galaxy before converging to a constant except for the constant offset, the fluxes in these columns value of 0.49 for r > 114′′. Galaxies often display such do not appear to be systematically affected. The three behavior, and it is not generally interpreted as a changing images in each filter were then shifted and coadded. We inclination angle with radius. Accordingly, we use an el- observed several Landolt (1992) standard fields for pho- lipticity of 0.49 for the whole galaxy. The corresponding tometric calibration, which was done with the IRAF im- inclination angle is 61.4◦, the same as the RC3 inclina- plementation of daophot (Stetson 1987). With the new tion of 61.5◦ within the uncertainties. The center of the standard stars in these fields identified by Stetson (2000) isophotal fits changed incoherently with radius before con- in addition to the original Landolt ones (we used Stetson’s verging for r > 120′′. The isophotal center was within a magnitudes for all of the stars), we had 34 - 38 standard few arcseconds of the visually obvious nucleus at (α, δ)= star measurements per filter. Our photometric solutions (09h47m15.3s, 67◦55m00.4s), so we used the nucleus as the were derived from a least-squares fit to the following for- fit center. The coordinates of the nucleus coincide with the mula: cataloged galaxy positions within their uncertainties (Cot- ton, Condon, & Arbizzani 1999; Falco et al. 1999). We m = minstr + C + f(V − I)+ g(a − 1) , (1) then ran ellipse again with all of the parameters fixed to produce the final surface brightness profiles. We fit ellipses where m is the apparent magnitude, m is ′′ ′′ instr every 2 out to a radius of 172 , where the ellipses began the instrumental magnitude (25 − 2.5 log flux + to run off the edge of the image. 2.5 log integration time), C is a constant that sets the We also ran ellipse on the Keck images with the same instrumental zero point, f is the color coefficient, V − I is parameters. This revealed some systematic differences the color of the object, and g is the extinction coefficient. between the Lowell and Keck photometry: although the Our observations did not span a large enough range of profile shapes were quite similar in the two datasets, the airmass to determine the extinction coefficient directly, Lowell V magnitudes are 0.1 mag brighter than the Keck so we used previously derived values. Reasonable ranges V magnitudes, and the Lowell I magnitudes are 0.1 mag for the coefficients are 0.2 < g < 0.4 mag airmass−1, B fainter than the Keck I magnitudes. The cause of this dis- 0.1 < g < 0.3 mag airmass−1, 0.05 < g < 0.15 mag V R crepancy is not clear, and it is worrisome because a 0.2 airmass−1, and 0.02 < g < 0.12 mag airmass−1 (P. I mag change in the galaxy color is significant. However, as Massey, private communication). For B and V, we used we will show in §3.1, the measured Lowell colors all pre- values of 0.27 mag airmass−1 and 0.15 mag airmass−1, dict stellar mass-to-light ratios that are consistent with which were the mean values of ∼ 15 measurements made one another, while the Keck V − I color predicts a notice- between 1997 and 1999 at the same telescope (D. Hunter, ably higher mass-to-light ratio that is inconsistent with the private communication). Lacking comparable measure- other determinations. An additional piece of evidence in ments in R and I, we used the standard Lick Observatory favor of the Lowell magnitudes is that the tabulated B −V values of 0.11 mag airmass−1 and 0.08 mag airmass−1, color in, e.g., the RC3 is close to our measured value, so respectively. These fall within the reasonable ranges for we conclude that it is safe to assume that our Lowell pho- both filters, and the Hunter B and V measurements are tometry is accurate. Since the LRIS field of view is larger very close to the Lick values. Since all of our images were than that of the Lowell CCD, we also used the Keck im- taken at airmasses close to 1.2, these assumptions are ages to verify that the light profile does not change shape unlikely to cause significant errors. at larger radii, and to measure the fraction of the total In order to double-check our photometric solutions, we flux that we missed due to the limited extent of the Lowell obtained the V- and I-band Keck4 images that Mendez mosaic. We estimate that ∼ 96% of the galaxy’s light is (2002) acquired for the purpose of measuring the TRGB contained within the r = 172′′ ellipse out to which we mea- distance to NGC 2976. These images were taken with the sured, so our integrated magnitude measurements should Low-Resolution Imaging Spectrometer (Oke et al. 1995), probably be revised upwards by 4 % (0.04 mag). and cover a 5′ × 7′ field. Exposure times were 300 s in I The measured surface brightness profiles are corrected and 400 s in V. by applying the Schlegel, Finkbeiner, & Davis (1998) Galactic extinction estimates in each band. To account 2.5. Surface Brightness Profiles for extinction within NGC 2976, we used an inclination- We used the IRAF routine ellipse in the STSDAS based approach, as described by Sakai et al. (2000). Sakai package to perform surface photometry on the images. et al. (2000) give internal extinction coefficients for all of The routine fits elliptical isophotes to a galaxy image at the bands we use except J, so to determine the J-band cor- specified radii, and allows the position angle (PA), elliptic- rection we interpolated their results from the other bands ity, and center to change with radius. There is no evidence and found that AJ = 0.8AI . For our best-fit axial ratio that the PA changes with radius, so we used the average 4 The W. M. Keck Observatory is operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. Dark Matter Halo of NGC 2976 5 of 1.96, we estimate that the internal extinction in mag- that is clearly not correct. We would therefore like to nitudes in the seven bands is (from B to Ks): 0.23, 0.20, have an independent constraint on the mass-to-light ratio 0.18, 0.13, 0.11, 0.07, and 0.03. so that we do not have to leave it as a free parameter. For The surface brightness profiles, displayed in Figure 5, this reason, and because we have BVRIJHKs photometry are qualitatively similar in all of the filters. NGC 2976 available, we choose the first method. clearly contains three components: a nucleus, an expo- nential inner disk, and an exponential outer disk. Since 3.1.1. Population Synthesis Constraints On M∗/L the nucleus is not resolved in any of our images, we used the HST/NICMOS images acquired by B¨oker et al. (1999) One way to estimate a stellar mass-to-light ratio from to estimate that its radius is less than 0′′. 36 (6 pc). It photometry alone is to use the semi-empirical relationships seems to be too reddened to reliably derive a mass-to- derived by Bell & de Jong (2001). Bell & de Jong (2001) light ratio from its colors. The nuclear luminosity is showed that the colors of spiral galaxies are strongly corre- 6 6 × 10 LK,⊙, so the rotation velocity due to the nucleus lated with the mass-to-light ratios of their stellar popula- 1/2 ′′ −1/2 −1 is 39(Mnuc/LK) (r/1 ) km s , where Mnuc/LK tions. With our multicolor photometry, we can construct is the stellar mass-to-light ratio of the nucleus in solar the entire array of colors for which they give formulas, and units. If we assume that the mass-to-light ratio is the then calculate the expected mass-to-light ratios, which are same as the maximum allowable value for the disk (see listed in Table 3. The average predicted values from the §4.2), the nucleus becomes dynamically insignificant out- six tabulated inner disk colors are 0.48 ± 0.02M⊙/L⊙K 5 side 10′′. Because the nucleus is probably a large cluster of in K band , and 1.07 ± 0.07M⊙/L⊙R in R band. That young stars (judging by its compactness, luminosity, and all of the colors predict consistent mass-to-light ratios is Hα emission), its actual mass-to-light ratio is likely much an indication that the predictions have some validity for lower. Parameters for the disk of NGC 2976 in each band this galaxy. It must be noted, however, that the Bell & de are listed in Table 1. The presence of an outer exponential Jong (2001) mass-to-light ratios are derived assuming that disk, with a surface brightness that declines more quickly galaxies have maximal disks. If the average galaxy has a than would be expected from extrapolating the inner disk, disk that is a factor f (0 ≤ f ≤ 1) less than maximal, then has been seen in other spiral galaxies (N¨aslund & J¨ors¨ater the predicted mass-to-light ratios from their calculations 1997; Pohlen 2001; Pohlen et al. 2002, hereafter PDLA). must also be scaled by the same factor f. A further uncer- The ratio of the inner scale length to the outer scale length tainty in this analysis is the initial mass function (IMF), is 2.1, consistent with the value of 2.0 ± 0.2 measured by which may not follow the assumed scaled Salpeter form, PDLA for four other galaxies. In fact, NGC 2976 only dif- particularly at low masses. fers from the galaxies PDLA observed in that the break be- An alternative method to measure M∗/LK photometri- tween the inner and outer disks occurs close-in, at 1 inner cally is to compare the observed colors directly to the out- disk scale length instead of ∼ 4 scale lengths. NGC 2976 puts of publicly available stellar population synthesis mod- is an order of magnitude less luminous than the galaxies in els. We used the Starburst99 population synthesis models the PDLA sample, suggesting that the break radius might (Leitherer et al. 1999) to attempt to constrain M∗/LK be a function of luminosity. in this way. For a given star formation history (constant star formation rate or instantaneous burst of star forma- 3. baryonic components of ngc 2976 tion), Starburst99 predicts colors and luminosities as a function of time. This allows us to search systematically Because our images of NGC 2976 do not reveal a bulge for the population age that matches the observed set of or a bar, and its nucleus is dynamically unimportant, the colors most closely. The two best Starburst99 models are only relevant reservoirs of baryons to consider are the stel- 1) a population with a small (10 %) young component lar and gaseous disks. that has been forming stars continously, and the remain- ing stars in an old (t & 3 Gyr) population that formed 3.1. The Stellar Disk in an instantaneous burst, and 2) a very young (∼ 107 yr There are two obvious approaches to studying the im- old) starburst. However, the mass-to-light ratios of these portance of the stellar contribution to a galaxy rotation models seem rather implausible: M∗/LK > 2 for model curve: 1) compare multicolor surface photometry of the 1) and M∗/LK ≈ 0.02 for model 2). Neither of these val- galaxy with the predictions of stellar population synthesis ues is compatible with observed values of M∗/LK in the models to obtain an estimate of the stellar mass-to-light few other galaxies for which measurements are available ratio (M∗/L) that is independent of the galaxy kinematics, (e.g., Olling & Merrifield 2001; Vallejo, Braine, & Baudry or 2) leave M∗/L as a free parameter while simultaneously 2002) or with the predictions of Bell & de Jong (2001). We fitting a scaled stellar disk and a dark matter halo to the also tried the online population synthesis code described observed rotation curve. The second technique has a very by Worthey (1994), which predicts colors and mass-to- 2 significant drawback: χ is insensitive to changes in M∗/L light ratios for an arbitrary combination of input stellar during rotation curve fits (McGaugh & de Blok 1998; Swa- ages and metallicities. It is more difficult to do a com- ters 1999; Bolatto et al. 2002), so the fit with the lowest prehensive search through the likely parameter space with value of χ2 does not necessarily convey any information this technique, but we did find that a mixture of 70 % of about the value of M∗/L. As an illustration of this effect, an old (but metal-rich) population and 30 % of a young the best fit often turns out to be M∗/L = 0, even though population (t = 1.1 Gyr) comes close to reproducing the 5 We will mostly use the K-band stellar disk for the remainder of the paper for the following reasons: 1) K-band light is the best tracer of the stellar mass distribution and the least skewed by luminous young stars from recent star formation, and 2) K band is the least affected by extinction. 6 Simon et al. observed colors (assuming a Miller-Scalo IMF), yielding a cover the large H II regions at either end of the inner disk, mass-to-light ratio of 0.31 M⊙/L⊙K. which are likely associated with molecular clouds. Never- We conclude that it is not possible to uniquely determine theless, the molecular material is not dynamically signifi- the M∗/LK for NGC 2976 by comparing the galaxy colors cant globally or locally, regardless of how it is distributed. 8 with the predictions of current population synthesis mod- The atomic gas mass is much larger, at 1.5 × 10 M⊙ (Ap- els. From this information alone, NGC 2976 could contain pleton et al. 1981; Stil & Israel 2002a). We adapt the H I either a very young starburst, or a normal, mixed stellar surface density distribution from the data presented by population with a low to moderate (0.3 to 0.5) M∗/LK. Stil & Israel (2002a). The stellar, atomic, and molecular There are three reasons for discounting the starburst pos- surface densities are plotted in Figure 6. Even with a low sibility in NGC 2976. First, the colors of the outer disk of M∗/LK, the stars are clearly the dominant reservoir of NGC 2976, where our observations show no evidence for baryons in NGC 2976. It is noteworthy that the H I and widespread star formation, are very similar to the inner stellar scale lengths in the outer galaxy appear nearly iden- disk colors, particularly in the near-infrared where extinc- tical, and the surface densities are comparable as well. We tion is less important (as shown in Table 3). It is unlikely calculate the rotation curves of the gaseous components di- that the galaxy contains a starburst and an old population rectly from their surface density profiles (again assuming that coincidentally have the same observed colors. Sec- zero thickness) using the same method as for the stars. ond, the starburst would have to be unusually young (in which case our observations of it at this particular time are 4. rotation curve and dark matter halo of rather fortuitous), and also quite strong, dominating not ngc 2976 only the light output from the galaxy, but also containing a significant fraction of its total stellar mass (otherwise Now that we have a handle on the behavior of the stellar the mass-to-light ratio would begin to run into the kine- and gas disks of the galaxy, we can move on to our primary matic limit; see §4.2). And finally, the visual appearance goal of constraining the structure of the dark matter halo. of the galaxy is not suggestive of a vigorous starburst. The First, we convert our two-dimensional velocity field into a more likely alternative is that NGC 2976 has a substantial one-dimensional rotation curve. This is accomplished by old component to its stellar population, driving M∗/LK fitting tilted-ring models to the velocity field using three toward the values of a few tenths that are seen in other complementary techniques. The algorithms are mentioned galaxies. briefly below, and more detailed descriptions are given in Appendix A. Rotcur breaks the velocity field into rings 3.1.2. Rotation Velocities Due to a Thin Disk and fits for the PA, inclination, center, systemic velocity, In order to compare the stellar rotation velocities to the and rotation velocity in each ring. Ringfit also divides observed rotation curve, we calculate the rotation veloci- the galaxy into rings, and it fits for the rotation velocity, ties for material confined to a thin disk. Because the disk the radial velocity (in the plane of the galaxy), and the sys- of NGC 2976 is not a pure exponential, its rotation curve temic velocity in each ring. Ringfit thus has the desirable must be calculated numerically. We perform the calcula- feature that a simple form of noncircular motions are in- tion with the routine ccdpot, which is based on a deriva- cluded in the fit. The third algorithm, rotcurshape, fits tion given by Binney & Tremaine (1987), in the NEMO the entire velocity field with a single PA, inclination, cen- package (Teuben 1995). This rotation curve is similar to ter, and systemic velocity, and also assumes a functional that from the fitted exponential disk out to the breakpoint form for the rotation curve and solves for the parameters between the inner and outer disks, where it briefly exceeds of that function. Fit results are similar for all three proce- the exponential disk rotation curve, and then begins to de- dures, although fitting for radial velocities in addition to cline more quickly (as expected, since the surface density rotation does make the rotation curve somewhat shallower at large radii is lower than in the single exponential case). (see §6.2). Our calculations assume an infinitely thin disk for sim- plicity; allowing the disk to have some thickness leaves the 4.1. Rotation Curve of NGC 2976 shape of the rotation curve almost unchanged, but lowers It is apparent from the data (Figure 4) that the velocity its amplitude, thus raising the allowed M∗/L (Swaters K field near the center of the galaxy cannot be adequately de- 1999; Peng et al. 2002). For a scale height equal to 1/6 scribed by rotation alone. There are two choices for how of the disk scale length, the rotation curve is lowered by to proceed: 1) use additional Fourier terms to describe about 10 % (Peng et al. 2002), so that the allowed M∗/L K the velocity field, or 2) allow for changes in the position may be 20 % higher than in the infinitely thin case (since angle of the galaxy with radius. The second possibility, vrot ∝ M∗/LK). which is difficult to reconcile with the photometry, is dis- cussed in §5.5; for now, we will use Fourier analysis to p The Gas Disk 3.2. provide an accurate description of the velocity field. The The atomic and molecular gas disks of NGC 2976 do next Fourier term beyond pure rotation (cos θ, where θ is not contribute appreciably to its measured rotation curve. the angle from the major axis in the plane of the galaxy; Although it is rich in CO for a dwarf galaxy, the measured see Appendix A) is pure radial motions (sin θ). We have total flux of ∼ 45Jykms−1 over the central 750 pc (Young investigated the decomposition of the velocity field using 7 et al. 1995) implies only ∼ 10 M⊙ of molecular gas (in- higher order terms, and found that they are much smaller cluding helium), if the Galactic CO-H2 conversion factor than the rotation and radial components and are consis- is valid in NGC 2976. The total molecular mass might be tent with noise. Therefore, we leave those terms out of our somewhat larger, because the BIMA observations did not subsequent analysis. Dark Matter Halo of NGC 2976 7

Our final rotation curve was derived with ringfit, so is beyond the scope of this paper. Nevertheless, we note that we could account for the radial motions that are that the radial motions are comparable in magnitude to present in the velocity field. We first fit the Hα and CO ve- the rotation only for the inner four points of the rotation locity fields separately to verify that they agree with each curve (the central 300 pc of the galaxy). At larger radii the other, as displayed in Figure 7a. At small radii (r < 40′′), rotation clearly dominates, and the orbits are nearly circu- it is evident that the derived rotation curves and radial lar. If we fit the rotation curve using only points between motions do agree, although they begin to diverge some- 300 pc and 1.8 kpc — where the radial motions are prob- what at r> 40′′. However, the CO ring fits at these radii ably unimportant — the derived density profile is almost are based on only one or two independent measurements, identical to the one described in the previous paragraph. so the apparent difference between the CO and Hα ve- This suggests that a more complete analysis of the effect of locity fields is not significant. Therefore, we combine the noncircular motions on the inversion from a rotation curve two datasets and fit again, weighting each data point by to a density profile should not have a large impact on the the inverse square of its statistical uncertainty. The fits derived slope of the density profile. for rings at r < 40′′ are displayed in Figure 8, where it is apparent that the velocity maxima do not lie along the 4.2. Limits on the Dark Matter Halo major axis. This indicates the presence of radial motions, To reveal the shape of the density profile of the dark which could not have been measured with longslit observa- matter halo, we first remove the rotational velocities con- tions or rotcur fitting. The rotation curve from these fits tributed by the baryonic components of the galaxy. The is plotted in Figure 7b, and the significance of the radial rotation curve of the dark matter halo is defined by velocities is again apparent. The estimated systematic un- certainties (the derivation of which is described in §6.2) are 2 2 2 2 2 vhalo = vrot − v∗,rot − vHI,rot − vH2,rot. (2) shown by the shaded gray areas surrounding each curve. The residual velocity field after subtracting this best- We determine the lower limit to the dark matter density fit model is displayed in Figure 9. Although individual profile slope by maximizing the rotation curve contribu- residuals are occasionally as large as 30 km s−1, the rms tion from the stellar disk. The maximum possible stellar of the residual field is 6.4 km s−1, and there are no obvi- rotation curve is set by scaling up the mass-to-light ratio ous systematic trends. The random velocity variations in of the stellar disk until the criterion the residual map are much larger than the uncertainties 2 2 2 2 −1 v < v − v − v (3) in the observed velocities, and the value of 6.4 km s is ∗,rot rot HI,rot H2,rot consistent with the random velocities of gas observed in is no longer met at every point of the rotation curve. other galaxies; the residuals therefore likely represent real This requirement sets maximum disk mass-to-light ra- +0.15 small-scale structure in the velocity field. The rotation tios of M∗/LK = 0.09−0.08M⊙/L⊙K and M∗/LR = velocities and radial velocities with their associated uncer- +0.56 0.53−0.46M⊙/L⊙R, where the uncertainties are calculated tainties, and the stellar and gas rotation curves, are all by replacing vrot with vrot ± δvrot in Equation 3. listed in Table 2. In order to incorporate more accurately We now use Equation 2 to obtain the rotation curve due the uncertainties in the rotation curve, the values listed to the dark halo. Under the assumption that the density −αDM in Table 2 and plotted in Figure 7b are the mean values profile can be described with a power law, ρDM ∝ r , obtained from a Monte Carlo study rather than directly we perform a linear fit to determine αDM as a function of § ′′ from the data (see 6.2.6). M∗/LK. The fit extends out to a radius of 105 , and we ig- The rotation curve of NGC 2976 is well-described by a nore points that have imaginary halo rotation velocities. A power law from the center of the galaxy out to a radius power law provides a good fit to the halo rotation curve for of almost 2 kpc, as displayed in Figure 10a. The resid- any mass-to-light ratio. The results of these fits are plot- uals after subtracting the fit from the rotation curve are ted in Figure 11. For M∗/LK > 0.19M⊙/L⊙K, αDM < 0 shown in the bottom panel. The rotation curve only be- and the density of the dark matter halo is increasing with gins to deviate systematically from power-law behavior at radius. Because such a dark matter configuration is prob- ≈ ′′ total r 110 (1.84 kpc). The (baryonic plus dark mat- ably unphysical, we consider 0.19 M⊙/L⊙K to be a firm ter) density profile corresponding to the rotation curve is upper limit to the stellar disk mass-to-light ratio, with the −0.27±0.09 −3 ρTOT = 1.6(r/1 pc) M⊙ pc (see Appendix B, corresponding lower limit to αDM of 0. The dark matter Equations B2 and B3, for the conversion between power density profile for this maximal disk is laws in velocity and density). This density profile is the mean of the fits to 1000 Monte Carlo rotation curves, −0.01±0.13 r −3 ρDM =0.1 M⊙ pc . (4) which represents a more accurate estimate of the uncer- 1 pc tainties than the fit to the single rotation curve shown   in Figure 7b. In the following subsection, we show that As we argued in §3.1.1, the only way that the stellar mass- the density profile of the dark matter halo alone follows a to-light ratio can be lower than this value is if the galaxy shallower power law. contains a young starburst, so Equation 4 represents the A key assumption underlying the derivation of this den- most likely shape for the dark matter halo. Note that even sity profile is that the orbits are circular, and therefore that though the kinematic value of M∗/LK we derive is rather the gravitational and centripetal forces are in equilibrium. low, there are two effects that we have not accounted for This assumption is not likely to be correct in detail, but that tend to raise it: the finite thickness of the stellar disk an inversion of the velocity field (including noncircular mo- (§3.1.2), and asymmetric drift (§6.2.7). Including these ef- tions) to obtain the underlying nonaxisymmetric potential fects raises the maximum disk M∗/LK close to the range that is predicted from the photometry. 8 Simon et al.

The slope of the total density profile of the galaxy rep- interesting to trace the rotation curve farther out as it pre- resents the absolute upper limit for the slope of the dark sumably flattens and turns over. We are in the process of matter density profile, so αDM ≤ 0.27 ± 0.09. In practice, using recently-obtained VLA H I observations to carry out because the galaxy contains stars and gas, the upper limit this study. must be lower. If NGC 2976 is not undergoing a strong Beyond numerical effects, though, there are more im- and very young starburst, its stellar mass-to-light ratio portant reasons to suspect that the simulations may not must be at least 0.10 M⊙/L⊙K. correspond well to the observations. One potentially sig- Therefore, we conclude that the dark matter density nificant problem with current simulations is that they ne- −0.17±0.09 0 profile is bracketed by ρDM ∝ r and ρDM ∝ r glect the effects of the baryons on the dark matter halo. (see Figure 11). Due to the extremely low value of the As we have shown, the central region of NGC 2976 is dom- maximal disk mass-to-light ratio, the galaxy must contain inated by the stellar disk. It is possible that the formation an essentially maximal disk. We adopt the M∗/LK = of a massive disk at the center of a cuspy spherical halo 0.19M⊙/L⊙K disk and αDM =0.01 halo, which are shown could destroy the central cusp (Weinberg & Katz 2002, in Figure 10b, as our preferred solution for the rest of the although see Gnedin & Zhao (2002)). paper. This disk dominates the gravitational potential of An additional possibility for accounting for the very the galaxy out to a radius of 35′′ (550 pc). The total mass shallow central density profile within the context of CDM of NGC 2976 out to the edge of the observed velocity field is suggested by the recent work of Stoehr et al. (2002) 9 at 2.2 kpc is 3.5 × 10 M⊙, of which 5 % is contributed by and Hayashi et al. (2003). These authors find in their the gas, and up to 14 % (for M∗/LK = 0.19M⊙/L⊙K) is simulations that dark matter satellite halos orbiting in contributed by the stars. the potential of a more massive neighbor are subject to tidal stripping. The stripped satellites end up with den- 5. discussion sity profiles that are much shallower than their original NFW profiles. If NGC 2976 can be identified with one of 5.1. Comparison to Cold Dark Matter Simulations the most massive few dark matter satellites of M 81, this In the previous section, we derived the dark matter den- mechanism provides a natural way to explain its nearly sity profile of NGC 2976, and determined that it cannot constant-density dark matter halo without modifying the have a slope steeper than αDM = αTOT =0.27±0.09. Even CDM model. in this minimum disk case an NFW halo in NGC 2976 is We conclude that the solution to the density profile ruled out. problem does not currently require fundamental changes to CDM. There are a number of simpler explanations that may remedy the discrepancy between observations 5.1.1. Does the Density Profile of NGC 2976 Conflict With CDM? and simulations. More complete simulations can help to clarify the situation, as can more carefully targeted high- The shallow central density profile of NGC 2976 does resolution observations (for example, studies of a few iso- not necessarily imply a problem with CDM. It is also pos- lated galaxies could confirm or refute the possibility that sible that the simulations and the observations may not density profiles are being modified by tidal stripping). be directly comparable, or that the simulations may not incorporate all of the relevant physics. Our observations 5.1.2. NFW and Pseudoisothermal Fits for NGC 2976 have only probed the very inner portion of the galaxy’s Up to this point, we have used power law fits to de- potential, whereas the numerical simulations are better at scribe the rotation curve and density profile, giving us a revealing the density structure at large radii. The highest- straightforward measurement of the central slope6. This resolution simulations can reach radii as small as 0.5 % of method has two advantages over the traditional approach the virial radius (Power et al. 2003). An NFW halo com- of fitting the rotation curve with various observationally parable in size to NGC 2976 would have a virial radius of or theoretically motivated functional forms to see which ∼ 80 kpc, so a simulation resolution element would be 400 one best matches the data. First, it is model-independent. pc in the best case. There would then be ∼ 5 resolution Second, some functional forms (NFW, for example) re- elements within the observed region of the galaxy, which quire that the data cover a certain range of radii in order might not be enough to accurately determine the slope of to constrain the fit parameters. An NFW rotation curve the density profile over those radii. It is therefore plau- reaches its maximum at 2.16rs and then turns over. If the sible that higher-resolution simulations could help to re- velocity data do not extend beyond the turnover radius, solve the apparent conflict between the observational and the scale radius (and hence the concentration parameter) theoretical results. It is also worth noting that none of of the halo cannot be reliably measured. the CDM simulations reported in the literature to date Although we argue that the power-law approach may have explored galaxies as small as NGC 2976. Although be more useful, we recognize that performing NFW and the simulated density profiles appear to be independent of pseudoisothermal fits to our data will facilitate compar- mass, simulated dwarf galaxies could conceivably have dif- isons to previous work. Accordingly, we have used the ferent density profiles than the large galaxies and galaxy rotcurshape routine (Appendix A.3) to attempt to find clusters that have thus far been studied. Finally, we point best-fitting parameters for the velocity field of NGC 2976, out that the current dataset just reaches what appears to assuming each of those functional forms for the rotation be the peak of the rotation curve; it would be extremely curve. An isothermal halo with a constant-density core 6 Note that in general a power law is not a good representation of the expected CDM density profile form, which has a logarithmic slope that varies from ∼−1 to −3. Our measurements, however, are all within the characteristic radius of the halo of NGC 2976, where the density profile predicted by CDM is close to a power law. Dark Matter Halo of NGC 2976 9 provides a reasonable fit to the data, with a core radius of is superficially rather similar to the first galaxy we stud- 67′′ (1.12 kpc) and an asymptotic velocity of 130 km s−1. ied, NGC 4605, so it is reasonable to compare the two. This fit is comparable in quality to the power law fit. For Our observations of NGC 4605 showed that its dark mat- an NFW rotation curve, rotcurshape cannot obtain a ter halo has a density profile with αDM =0.65 (Bolatto et satisfactory fit for any value of the concentration. We also al. 2002). At first glance, this result does not appear to attempted to fit the NFW form just to the rotation curve seriously conflict with our findings for NGC 2976. How- (not the full velocity field) with various nonlinear least- ever, the NGC 4605 density profile was for a maximal disk, squares techniques. Because we know that the rotation and therefore represents a lower limit on αDM. We argued curve of NGC 2976 is shallower than an NFW rotation that the maximum disk solution was the most likely for curve, we fixed the concentration parameter at an artifi- NGC 4605 because the mass-to-light ratio could not real- cially low value (c = 9.2, ∼ 2σ lower than expected; Bul- istically be much smaller than its maximum value of 0.22 lock et al. (2001)) for these fits, and only solved for v200 M⊙/L⊙K in that galaxy, and because it leads to a simpler and r200. We found that neither v200 nor r200 are signif- density structure for the halo (a single power law rather icantly constrained by the rotation curve of NGC 2976. than two). The best NFW fits have a reduced χ2 value of 6.2 (com- For NGC 2976, by contrast, we set an upper limit of 2 pare to a reduced χ of 1.3 for a power law fit), and the αDM = 0.27 for the minimum disk case, and we prefer NFW rotation curve only passes within 1σ (combined sta- lower values of αDM because a minimum disk is not phys- tistical and systematic uncertainties) of 2 out of the 27 ically realistic. For solutions in the range that we believe points in the rotation curve. The remainder of the fitted is reasonable (0 ≤ αDM ≤ 0.17), the dark matter density points are up to 4.1σ away from the data points, show- profile slope disagrees with that of NGC 4605 by up to ing that an NFW rotation curve is very strongly excluded 5σ, even though the disks of these two galaxies are quite for this galaxy. Note that both the pseudoisothermal and similar. Although we have only examined two galaxies so NFW fits described here were performed on the total mass far, their incompatible dark matter density profiles suggest distribution of the galaxy, not just the contribution from that the cosmic scatter in halo properties may be large. the dark matter halo. Removing the stellar and gas disk velocities first would make the NFW fit worse. 5.3. Are All Dwarf Galaxies Dark-Matter Dominated? Although our velocity field does not extend beyond the turnover of the rotation curve and NFW fits to the rotation It is generally assumed that, with the possible exception curve are unconstrained, there is another way to estimate of tidal dwarfs (Barnes & Hernquist 1992), all dwarf galax- the NFW concentration parameter, and the effective con- ies are dynamically dominated by dark matter (Carig- centration parameters for other dark halo models from the nan & Freeman 1988; Carignan & Beaulieu 1989; Jobin data. Alam, Bullock, & Weinberg (2002) defined the pa- & Carignan 1990; Martimbeau, Carignan, & Roy 1994). While this assumption is likely true for the outer parts of rameters RV/2 (the radius at which the rotation curve has risen to half of its maximum value) and ∆ (the mean dwarfs (radii larger than ∼ 2 times the disk scale length), V/2 the observational evidence is more ambiguous close to their density within R , in units of the critical density) in V/2 centers. One of the main sources of this problem is that order to make it easier to compare rotation curve observa- dwarf rotation curves are traditionally observed in H I, tions with theoretical predictions. For the minimum disk ′′ −1 with angular resolution as low as 30 . The rotation curves case in NGC 2976, Vmax =86kms , RV/2 = 768 pc and 6 therefore often contain only two or three data points at ∆V/2 = 1.3 × 10 . Using the formulae given by Alam et radii where the stellar and gas disks are dynamically im- al. (2002), we calculate concentrations of 18.5, 4.1, 51.5, portant. To make matters worse, these inner data points and 225.4 for an NFW profile, a Moore profile, a Burk- are the most likely to be affected by beam smearing and ert profile, and an isothermal+core profile, respectively. other systematic problems. We suggest that higher reso- For our preferred solution, after accounting for the stel- lution observations of dwarf galaxies may show that their lar and gas disks, the dark matter halo parameters are −1 5 central regions are often dominated by luminous material. Vmax =74 km s , RV/2 = 902 pc and ∆V/2 =7.0 × 10 , In the case of NGC 2976, the baryonic mass dominates reducing the concentrations to 14.5, 3.1, 40.8, and 165.0. the central 220 pc of the galaxy even for the lower limit The Alam et al. (2002) analysis is designed to study to M∗/LK of 0.10 M⊙/L⊙K. For our preferred solution of the value of the central density of the dark matter halo a maximal disk (M∗/LK =0.19M⊙/L⊙K), the disk dom- (which is also a potential point of disagreement between inates out to a radius of 550 pc. Consequently, the stellar observations and simulations). With or without account- disk has a significant impact on the derived density profile ing for the baryonic contribution to the rotation curve, of the dark matter halo: slopes ranging from αDM = 0.29 the central density of the dark matter halo of NGC 2976 to αDM = −0.13 are possible depending on the choice of (parameterized by Alam et al.’s definition of ∆ ) is con- V/2 M∗/LK (see Figure 11). sistent with ΛCDM simulations, even though the shape of That stars contribute to the dynamics of a dwarf galaxy the density profile is not. is not unique to NGC 2976; similar conclusions were reached for NGC 1560 by Broeils (1992), for NGC 5585 by Comparison to NGC 4605 5.2. Blais-Ouellette et al. (1999), and for NGC 4605 by Bolatto NFW suggested, and most subsequent authors have et al. (2002). In addition, this result is also in agreement agreed, that relaxed CDM halos should all have the same with the work of, e.g., Persic, Salucci, & Stel (1996, here- shape independent of mass or merger history7. NGC 2976 after PSS), who showed that the fraction of dark mass in 7 Provided that they have not recently undergone a major merger. There is no kinematic or photometric evidence to suggest that either of the galaxies discussed here was recently involved in a merger. 10 Simon et al.

−0.56 spiral galaxies is a strong inverse function of luminosity. obtained under these assumptions is ρTOT ∝ r . Since PSS found that in galaxies with luminosities comparable the photometric PA of the galaxy is quite stable, varying to NGC 2976 (MI = −18.5), dark matter can be detected only a few degrees from its average value beyond a ra- gravitationally beginning at 10 − 15% of the optical ra- dius of 30′′ (at smaller radii, local bright spots dominate dius (which is located at 2.8 kpc for NGC 2976), or about the isophotal fitting), this model requires a physical mech- 350 pc. This is entirely consistent with our mass modeling anism that could cause the behavior of the photometric (see Figure 10b). The average MI = −18.5 rotation curve and kinematic PAs to deviate strongly from one another. constructed by PSS has dark matter dominating the rota- It is unclear what such a mechanism could be, and why it tion curve at radii beyond 0.2Ropt (560 pc), also consistent would make the kinematic PA change so rapidly. Because with our preferred solution. Thus, even though it may this model lacks an observational motivation, while radial seem counterintuitive, the PSS results support our con- (or other noncircular) motions are expected to occur nat- clusion that luminous matter is sometimes an important urally for a variety of reasons (see below), we prefer the contributor to the inner rotation curves of dwarf galaxies. radial motion interpretation of the velocity field. There are a number of possible sources of the radial 5.4. Are the Kinematics of NGC 2976 Affected By M 81? motions. The galaxy could contain a stellar bar, although NGC 2976 does not appear to be participating in the there is no sign of a bar in any of our images, even at 2.2 dramatic tidal interaction currently taking place between µm. Further evidence against the presence of a bar is the M 81, M 82, and NGC 3077 (Yun, Ho, & Lo 1994); how- lack of measurable higher order terms in our harmonic de- ever, it has likely interacted with M 81 in the past. Ap- composition of the velocity field. The velocity field of a pleton, Davies, & Stephenson (1981) discovered a faint H I barred galaxy should contain a nonzero sin 3θ term (Wong streamer stretching from M 81 to NGC 2976. Boyce et al. 2000). An alternative to a bar is the possibility that the (2001) used HIJASS data to show that this gas comprises a dark halo is triaxial rather than spherical, as we have as- single tidal bridge that smoothly connects the two galaxies sumed. It is expected that CDM halos should be at least (see their Figure 2a). The bridge contains somewhat more moderately triaxial (Dubinski & Carlberg 1991; Warren et 8 8 al. 1992; Cole & Lacey 1996), and the potential of a triaxial H I than NGC 2976 itself (2.1 × 10 M⊙ and 1.5 × 10 halo is certainly not axisymmetric, so the velocity field of M⊙, respectively). Unfortunately, the HIJASS observa- tions lack the angular resolution to see the details of the a galaxy embedded in a triaxial halo would exhibit noncir- connection between the bridge and NGC 2976, and the cular motions. However, since the details of such a veloc- presence of Galactic H I further complicates the situation. ity field have not yet been simulated, we cannot compare Yun, Ho, & Lo (2000) suggested that the bridge is a rem- our results to theoretical predictions. Future simulations nant of an interaction that took place only between M 81 of the kinematics of a gaseous disk within a triaxial halo and NGC 2976 before the current M 81/M 82/NGC 3077 would be quite interesting. Other potential causes of the event. Nevertheless, the optical galaxy (Figure 1) and the radial motions include a disk that has nonzero ellipticity, inner H I disk (Stil & Israel 2002a,b) both appear regu- and outflows associated with star formation. lar, symmetric, and undisturbed. Assuming that M 81 12 has a total mass of ∼ 10 M⊙ (Karachentsev et al. 2002), 6. systematics its tidal field only becomes comparable to the gravity of In this section, we study in detail the systematic uncer- NGC 2976 (at a radius of 2 kpc) if the galaxies approach tainties in our analysis, and also some systematic problems within 20 kpc of each other. Since M 81 is currently at a that afflict rotation curve studies in general. We emphasize projected distance of 79 kpc, the present-day kinematics that systematic effects are the dominant source of uncer- of NGC 2976 are probably unaffected by the interaction. tainties in our analysis. Some of the details contained in this section are therefore crucial to understanding the re- 5.5. Possible Origins of Noncircular Motions liability of our conclusions. The general reader may wish In Figure 4a it is clear that the velocity field of to use the summary in the following paragraph and the NGC 2976 is distorted compared to a purely rotating disk. subsection headings to select the portions in which he or The velocity gradient near the center of the galaxy is not she is interested. directed along the photometric major axis, but is offset by We begin in §6.1 by mentioning the importance of con- up to ∼ 40◦ (see Figure 8). This twisting of the isove- sidering systematic problems, and our efforts to account locity contours means that the kinematics of NGC 2976 for these problems in the design of our survey. Section cannot be described by the simplest model: a constant 6.2 continues with a description of our tests for system- PA and only rotational motions. We have shown that the atic errors caused by the rotation curve fitting. In §6.3 velocity field can be adequately described by adding ra- we demonstrate that the Hα and CO velocity fields of dial motions in the plane of the galaxy to the model. If NGC 2976 are consistent with each other, not just globally, there are systematic trends remaining after this model has but on a point-to-point basis. In §6.4 we use our velocity been subtracted from the data, they are only present at field to simulate longslit observations of NGC 2976, and the level of a few km s−1 (see Figure 9). However, a purely compare the derived longslit density profiles to the one rotational velocity field with a kinematic PA that declines we extract from the two-dimensional velocity data. Sec- monotonically from ∼ 6◦ near the center of the galaxy to tion 6.5 examines the problem of offsets between the kine- −37◦ at a radius of 90′′, and remains constant at −37◦ for matic center of a galaxy and the position of the slit during larger radii can also fit the data. This model is the one spectroscopic observations, and §6.6 briefly discusses the produced by rotcur if the kinematic PA is left as a free difficulties that barred galaxies present for density profile parameter (see Appendix A.1). The total density profile studies. Dark Matter Halo of NGC 2976 11

6.1. The Problem of Systematics weighting, and no radial motions) and considering radii ′′ It is well-known, although not often discussed, that less than 105 , the algorithms all produce essentially iden- tical results. We conclude that none of the assumptions there are a number of serious systematic uncertainties that that are built in to the fitting algorithms affect the results. can cause an observed rotation curve (and the associated density profile) to differ significantly from the true one. The only significant difference that appears between the algorithms stems from the inclusion of radial velocities in Worse, nearly all of these effects work in the same di- the fit. Earlier, we noted that it is obvious from inspection rection to cause density profiles to appear systematically shallower than they actually are. Fortunately, the most of the velocity field (Figure 4) that noncircular motions are present in NGC 2976: for example, the velocity fits in severe of these problems can be minimized or avoided by individual rings for r< 40′′ show that the observed veloc- using two-dimensional velocity fields and by making veloc- ity measurements at very high precision (Blais-Ouellette ity maximum is systematically offset from the photometric major axis (Figure 8). Neglecting the sin θ term and fitting et al. 1999; van den Bosch & Swaters 2001; Swaters et only for rotation increases the exponent of the density pro- al. 2002; Bolatto et al. 2002). SMVB model several ′′ of these effects in detail and determine how severely ob- file from αTOT =0.27 to αTOT =0.42 (for 0

6.2.5. Uncertainty in Systemic Velocity 6.2.8. Conclusions From Analysis of Rotation Curve Systematics If the systemic velocities are left as a free parameter in the velocity field fits, they have a scatter of 1.8 km s−1 We have shown that the only ways to significantly from ring to ring. It is possible that these variations are change the derived slope of the density profile of NGC 2976 real, although they are only marginally significant when are to 1) assume the stellar mass-to-light ratio is zero, 2) the systematic uncertainties are taken into account. The ignore the radial component of the velocity field, or 3) al- alternative approach is to fix vsys at its average value and low the kinematic PA and/or inclination to change with then fit again with only vrot and vrad as free parameters. radius. Assuming that the observed velocities are due en- TOT ∼ The results of the fit with vsys fixed are nearly identical tirely to rotation raises α by 0.15, and allowing the to the previous results. None of the radial or rotational PA and inclination to change with radius raises αTOT by velocities are changed by more than 1σ, the density pro- up to an additional ∼ 0.25. Accounting for the contribu- file exponent for the total mass distribution increases by tion of the maximum stellar disk, however, limits the dark ≤ DM ≤ less than 1σ (to αTOT = 0.34 ± 0.09), and the maximum matter density profile exponent to 0.26 α 0.4. allowed mass-to-light ratio increases to 0.24 M⊙/L⊙K. Because inspection of the velocity field and the fits clearly reveals the presence of radial motions, neglecting 6.2.6. Uncertainties in Rotation Velocities and Radial the radial component is not justified. Changes in the PA Velocities with radius are not supported by the photometry, and changes in the inclination with radius are difficult to un- Using the measured uncertainties in the center position derstand physically. Therefore, we argue that these solu- ◦ and PA, and assuming an uncertainty of 3 for the in- tions, despite being mathematically viable, are contrived clination angle, we calculated the resulting uncertainties and not motivated by the data. on the rotation velocities and the radial velocities with a We conclude that the galaxy contains substantial radial Monte Carlo technique. We generated 1000 random cen- motions, and that the density profile results are not signif- ters, PAs, and inclinations, assuming a Gaussian distri- icantly affected by the most obvious sources of systematic bution for each of the parameters, and ran ringfit with errors. We caution that the robustness against systemat- each set of parameters. The standard deviation of the 1000 ics that we find is specific to this dataset, and may not be rotation velocities in each ring was defined to be the sys- true in general. Because the rotation curve of NGC 2976 tematic error of that rotation velocity, and the systematic increases with radius so slowly, errors in any of the geomet- errors in the radial velocities and systemic velocities were ric parameters of the galaxy are diminished in importance. calculated in the same way. The systematic errors on the A galaxy with a more rapidly rising rotation curve would − − rotation curve range from 2.1 km s 1 to 5.5 km s 1, as probably be more severely affected. Assuming that the ra- listed in Table 2. Power law fits to the 1000 Monte Carlo dial motions provide no support, the dark matter density rotation curves yield a mean slope of the total density pro- profile slope is in the range 0 ≤ αDM ≤ 0.27, with a 2σ file of αTOT =0.27 ± 0.09. upper limit of αDM ≤ 0.45, where systematic errors have been included in the uncertainty on αDM. NGC 2976 thus 6.2.7. Asymmetric Drift Correction violates the prediction of universal central density cusps We have also calculated the asymmetric drift correction by CDM simulations. to the rotation curve, as defined by, e.g., Cˆot´e, Carignan, 6.3. Comparing Velocities Derived From Different & Freeman (2000). We derived the velocity dispersion σ Tracers as a function of radius from the Hα data, and the surface density Σ by adding the H I and H2 column densities. We Some recent studies in the literature have shown that, then fit polynomials to σ(r) and Σ(r) and calculated the beam smearing questions aside, there do not appear to be derivatives dσ/d ln r and d ln Σ/d ln r analytically. There systematic offsets between H I and Hα rotation velocities are significant uncertainties that factor into this calcula- (e.g., McGaugh, Rubin, & de Blok 2001; Marchesini et al. tion, including 1) we have not included the ionized gas 2002). With a handful of exceptions, though, these studies surface density (although its contribution is expected to employed longslit Hα data, so the comparisons essentially be small), 2) the H2 surface density is uncertain due to took place only along the major axis. In addition, the spa- our imprecise knowledge of the CO-H2 conversion factor, tial and velocity resolution of the H I and Hα data were 3) our velocity field extends to radii that are smaller than often quite different. Dark Matter Halo of NGC 2976 13

In this paper, we have presented for the first time the the correlation coefficient of the two sides, the rms dif- data necessary for a two-dimensional comparison across a ference in velocity between points at the same radius on dwarf galaxy of the CO and Hα velocity fields. The an- opposite sides, and the appearance of the rotation curve. gular resolution of the two datasets is similar (6′′ and 4′′, These criteria are combined in a necessarily somewhat sub- respectively), and although the CO velocity resolution is jective manner, but since we know the true center in this better by a factor of ∼ 6, the higher signal-to-noise at Hα case from our two-dimensional velocity field, we have ver- compensates such that the velocities can be measured with ified that the chosen center never differs from the actual comparable precision. We use the following technique to one by more than 12′′ (200 pc). We fold the rotation compare the velocity fields. At the position of each Hα curve about the chosen center and average the two sides fiber, we compute a weighted average of the velocities of together, weighting each point by the inverse square of its all of the pixels in the CO map that fall within the radius uncertainty. Finally, we fit a power law to the resulting of the fiber. CO pixels that do not contain any emission rotation curve, ignoring any points near the center that are not used in computing the average, and of course, Hα have negative rotation velocities. We repeat this process fibers that do not coincide with any molecular emission are with offsets from the major axis of up to 14′′ (230 pc). not used either. This process yields a unique one-to-one The indices, αTOT, of the power law fits in density for each mapping between the two velocity fields. The rms differ- rotation curve are displayed in Figure 13. −1 ence between vHα and vCO is 5.3 km s , with the com- The naive expectation from this experiment is that slits parison being made at 173 points. Similar studies in the placed off of the major axis will make the density profile Milky Way found that the dispersion between the veloc- appear to be shallower than it actually is, and that this ities of molecular clouds and the associated Hα-emitting effect should become more severe with increasing distance gas was 4-6 km s−1 (Fich, Treffers, & Blitz 1982; Fich, from the major axis. The actual results do not show this Dahl, & Treffers 1990), so much of the scatter we observe trend very clearly. The positive slit offsets (corresponding in NGC 2976 may be intrinsic to the process of H II region to the northeast side of the galaxy) appear to agree with formation rather than caused by observational uncertain- the expected behavior; for large slit offsets the slopes are ties. We plot the Hα velocities against the CO velocities on average shallower than the value that should be derived in Figure 12. There is a weak systematic trend visible in (αTOT =0.42, since we are neglecting radial motions). Off- the residuals, with vCO > vHα near the center of the galaxy sets on the other side of the galaxy, though, do not follow and vCO < vHα on the northwest side of the galaxy. The a systematic trend. The derived slopes for negative off- amplitude of this trend is only a few km s−1, so it does not sets are similar to the actual slope. Note that when the affect our rotation curve. The origin of the trend is not rotation curve is folded about the correct central position clear, but we suggest that it could be a result of the spatial (instead of the one that gives the most symmetric appear- distribution of the gas. For example, where the ionized gas ance), the slopes are systematically shallower than when is largely in front of the molecular clouds, the expansion other central positions are used. We speculate that this of H II regions away from nearby molecular clouds would systematic effect is not very strong in NGC 2976 because make vHα > vCO. This effect should appear preferentially this galaxy has a relatively shallow central velocity gra- where Hα emission is bright. Conversely, where the molec- dient. Other galaxies with steeper rotation curves might ular clouds are in front, one would expect that vHα < vCO. be affected more severely. A possible explanation for the These areas should have faint Hα emission due to extinc- difference between the two sides of the galaxy is that the tion within the molecular clouds. The Hα distribution in Hα distribution is rather asymmetric; to the northeast of NGC 2976 appears to be qualitatively consistent with this the major axis are a number of bright point sources, while interpretation; the Hα is brighter in the northwest, where the emission on the southwest side is faint and diffuse. In the Hα velocities are larger, and there is an Hα hole to addition to coherent systematic errors, this exercise shows the southeast, where some of the CO velocities are higher. that attempting to derive a density profile from a single In any case, we conclude that the Hα and CO velocity velocity cut through a galaxy is also a noisy process. De- fields agree, with a scatter of 5.3 km s−1, and thus that pending on the position of the slit, one could estimate a −0.13 −0.82 both species should be accurate tracers of the gravitational density profile between ρTOT ∝ r and ρTOT ∝ r potential of NGC 2976. for this galaxy. Only with observations along many slits, or full two-dimensional velocity data, can one be confident 6.4. Simulated Longslit Observations of NGC 2976 that the rotation curve and density profile of a galaxy ac- curately reflect its gravitational potential. Our Hα dataset is well suited for studying the sys- tematic problems associated with deriving rotation curves 6.5. Positioning Errors and Slit Offsets from longslit spectroscopy. It is straightforward to recreate what would be seen by an observer taking longslit spectra There are several factors that can play a role in position- of NGC 2976. We begin by selecting all of the fibers within ing errors. First is the pointing and guiding of the tele- 1′′ of a given cut parallel to the major axis of the galaxy. scope used to acquire the data. McGaugh et al. (2001) and This creates an unevenly-sampled rotation curve, which de Blok & Bosma (2002) report that observations of LSB we smooth by averaging the points into 4′′-wide bins. We galaxies with different telescopes and instruments by in- then proceed exactly as we would if we had obtained these dependent observers result in essentially identical rotation rotation velocities from a longslit spectrograph. We find curves. On this basis, they conclude that pointing errors the center of the rotation curve by folding it about various do not impact their results. Telescope pointing and guid- points to determine the position of maximum symmetry. ing are thus unlikely to cause problems for longslit obser- Three criteria are used to judge the degree of symmetry: vations, although they can be an issue for two-dimensional 14 Simon et al. observations like ours, where the galaxy may not be vis- result in similar density profiles. However, one of the three ible on the guiding camera while observing it (see §2.2 slopes they measure (α = 0.32, −0.16, 0.30) differs from for our solution to this problem). Quite independent of the others by 3σ, showing that slit offsets can cause den- the telescope pointing, though, is the question of whether sity profiles different from the true one to be derived. Also, the center of a given galaxy is where it is reported to be. if none of the three slits went through the actual center of Even when the photometric center is known accurately, the galaxy, then further observations may be needed to galaxies can have dynamical centers that differ from the ensure that the measured density profile is correct. Given photometric ones by hundreds of parsecs (Puche, Carig- this degree of confusion over the position of a relatively nan, & Wainscoat 1991; Helfer & Blitz 1995; Matthews & nearby (d = 10 Mpc) and high-surface brightness (for a −2 Gallagher 2002). Because of these two additional prob- galaxy classified as LSB; µR = 21.6 mag arcsec ) galaxy, lems, a demonstration that pointing errors are minimal it is certainly not obvious that the centers of fainter and does not suffice to prove that rotation curves are system- more distant galaxies are well-determined in the literature. atically unaffected by offsets between their assumed and actual centers. 6.6. Barred And Highly Inclined Galaxies One example in the literature of a galaxy in which a Two other common attributes of galaxies that can cause poorly-determined center may have caused erroneous con- systematic errors deserve mention here: bars and high in- clusions about its density profile is NGC 2552, often re- clination angles. SMVB already discussed the problems ferred to by its alternate name of UGC 4325. The rota- associated with galaxies that are seen edge-on or nearly tion curve of this dwarf LSB galaxy has been discussed in a so and demonstrated that observations of such galaxies number of recent papers (van den Bosch & Swaters 2001; must be analyzed with extreme care. Equally problem- de Blok et al. 2001; Swaters et al. 2002; de Blok et al. atic, though, are galaxies that contain bars. Barred galax- 2002; Marchesini et al. 2002; de Blok & Bosma 2002). All ies certainly have noncircular motions out to the end of of these authors find density profiles with α ≈ 0.3, where the bar, so one-dimensional velocity data will be system- α is the central slope of the density profile (see §4.2 or atically affected. Density profiles of barred galaxies de- Equation B2). With six independent analyses reaching the rived from longslit data are therefore not trustworthy at same conclusion, it would seem that the density profile of radii less than the bar semimajor axis. SMVB include five this galaxy is well-determined. However, closer inspection barred galaxies in their sample, and unsurprisingly find reveals a potentially important discrepancy between these that these objects have shallower central density slopes studies: they assume widely varying central positions for than their other targets. The kinematics of barred galax- the galaxy (see Table 4). Two papers (van den Bosch & ies are interesting in their own right; there are suggestions Swaters 2001; Swaters et al. 2002) measure the galaxy’s that the presence of a bar can affect the evolution of dark center from their own photometry, while the other four matter density cusps (Weinberg & Katz 2002), and with make no reference to the assumed center. It is reasonable two-dimensional velocity fields it is possible to use barred to suppose that they used the coordinates given by one galaxies to study dark matter density profiles (Weiner et of the standard databases, such as the NASA/IPAC Ex- al. 2001; Weiner, Sellwood, & Williams 2001). However, tragalactic Database9 or the SIMBAD Database10. These ′′ longslit observations of barred galaxies may not be a reli- different positions span a range of 11 , or 550 pc at the able way to attack the density profile question. distance of NGC 2552. The reason for the uncertainty in the galaxy center is 7. conclusions clear from inspection of a Digitized Sky Survey image: We have used two-dimensional velocity fields, sampled NGC 2552 is a lopsided galaxy, with a low surface bright- at high spatial resolution and high spectral resolution in ness outer disk that is off center relative to the brighter CO and Hα to study the density profile of NGC 2976 and inner disk. However, there has recently been an accurate the parameters of its stellar disk and dark matter halo. We determination of the actual position of the galaxy; B¨oker, obtained rotation curves from the two-dimensional data Stanek, & van der Marel (2003) used HST imaging to show using tilted-ring models derived with three independent that NGC 2552 contains a nuclear star cluster, and that and complementary algorithms. Our tilted-ring fitting this nucleus is also the center of the inner isophotes. Be- shows that there are significant radial (i.e., noncircular) cause galaxies can display offsets between their nuclei and motions in the inner 20′′ (300 pc) of the galaxy. Ac- their dynamical centers, it is possible that the nucleus is counting for these motions yields a total density profile not located at the dynamical center of NGC 2552, but in −0.27±0.09 of ρTOT ∝ r . There is a narrow range of possible the absence of two-dimensional kinematic information it stellar mass-to-light ratios for NGC 2976, and the cor- represents the best guess. As can be seen in Table 4, the ′′ responding dark matter halo density profiles range from previously-used positions are up to 9 (450 pc) away from −0.17±0.08 −0.01±0.13 ρDM ∝ r to ρDM ∝ r (constant den- the nucleus. Based on the results of SMVB and our discus- sity). A key assumption that we make in the inversion of sion in Section 6.4, we suggest that density profiles derived the rotation curve to obtain the density profile is that the from the longslit observations cited above could have been gravitational and centripetal forces are in equilibrium (or significantly over or underestimated. de Blok et al. (2002) equivalently, that the radial motions provide no support). argue that this is not the case because the three slit posi- The density profile obtained by excluding measurements tions they used to observe NGC 2552 (one through their inside the 20′′ radius is identical to that computed when assumed center, and the other two 5′′ away on either side) including them, substantiating this assumption. 9 http://nedwww.ipac.caltech.edu/ 10 http://simbad.u-strasbg.fr/ Dark Matter Halo of NGC 2976 15

We found that in our preferred model, the maxi- following up on the recent work of SMVB and de Blok et mum mass-to-light ratio of the stellar disk of NGC 2976 al. (2002). We found that systematic errors can cause the +0.15 is M∗/LK = 0.09−0.08M⊙/L⊙K. If M∗/LK > density profiles inferred from longslit observations to differ 0.19M⊙/L⊙K, the dark matter halo has the unphysical significantly from the true density profiles. We also illus- property that its density increases with radius. Account- trated the difficulties that can arise in determining the po- ing for the thickness of the stellar disk and the asym- sitions of galaxy centers without adequate two-dimensional metric drift correction to the rotation curve brings this kinematic and photometric information. These problems kinematic value of M∗/LK into line with photometric esti- — as well as the disk-halo degeneracy — can be largely mates. Comparison with stellar population synthesis mod- overcome by using high-resolution two-dimensional veloc- els (Worthey 1994; Leitherer et al. 1999; Bell & de Jong ity fields, as we have shown in this paper. 2001) suggests that the mass-to-light ratio is unlikely to Although previous studies have found that central den- be less than 0.10 M⊙/L⊙K, so the stellar disk — and sity cusps cannot be ruled out in many dwarf and LSB hence the dark matter halo — are tightly constrained. We galaxies, we have demonstrated that a cusp is not present investigated many of the likely systematic effects on the in NGC 2976; the dark matter halo of this galaxy is nearly rotation curve and found that none of them can bring the constant density out to the edge of the observed Hα emis- −1 density profile close to ρDM ∝ r . We therefore rule out sion at a radius of 2.2 kpc. an NFW or Moore et al. (1999b) density profile in the center of this galaxy at high confidence regardless of the stellar contribution. This research was supported by NSF grant AST- In addition, we investigated the most extreme models 9981308. We thank the referee, Rob Swaters, for sug- of the galaxy that are allowed by the data. Density pro- gestions that improved the paper. In addition, we would file slopes as high as αDM ∼ 0.7 can be obtained, but only like to thank Di Harmer for her assistance both with when all three of the following are true: 1) the mass-to- the preparation of our observing proposal and with us- light ratio of the matter in the disk is zero, 2) the observed ing DensePak, and we acknowledge the telescope operat- velocities are attributed entirely to rotation, despite the ing skills of Gene McDougall and Hillary Mathis. We also observed radial motions, and 3) the kinematic PA and in- thank Amanda Bosh for her help with our Lowell observ- clination both change with radius in the manner described ing. JDS gratefully acknowledges the invaluable assistance in §5.5 and §6.2.8. Retaining requirements 2 and 3, but of Peter Teuben in getting rotcur up and running and assuming a maximal stellar disk, reduces αDM to . 0.4. then modifying the code to better suit our needs. This We consider these models to be quite unlikely, and incon- publication makes use of data products from the Two sistent with the complementary data we have presented Micron All Sky Survey, which is a joint project of the for this galaxy. University of Massachusetts and the Infrared Processing We also discussed whether a universal dark matter halo and Analysis Center/California Institute of Technology, shape is consistent with our observations. In a similar funded by the National Aeronautics and Space Admin- study of the slightly more massive galaxy NGC 4605, Bo- istration and the National Science Foundation. This re- latto et al. (2002) found a lower limit to the dark matter search has also made use of the NASA/IPAC Extragalactic density profile slope of αDM =0.65. Since the upper limit Database (NED) which is operated by the Jet Propulsion for NGC 2976 is αDM =0.27, the density profiles of the ha- Laboratory, California Institute of Technology, under con- los of these galaxies are different from one another. If the tract with the National Aeronautics and Space Adminis- disk of NGC 4605 is submaximal, or the disk of NGC 2976 tration, NASA’s Astrophysics Data System Bibliographic is not minimal (which is likely), the inconsistency becomes Services, the SIMBAD database, operated at CDS, Stras- more severe. In addition, we note that both of these dwarf bourg, France, and the LEDA database (http://leda.univ- galaxies are dynamically dominated by luminous matter lyon1.fr). Finally, we would like to thank Wendy and Lil- at small radii. iana for allowing us to cruelly abandon them in order to Finally, we considered the impact of some of the known observe in such faraway places as Arizona and northern systematic uncertainties that afflict rotation curve studies, California.

APPENDIX A. rotation curve fitting algorithms When dealing with longslit kinematic data it is relatively straightforward to transform the reduced observations into a rotation curve. For a full velocity field, the process is more complicated because it involves converting two-dimensional data to one dimension while retaining as much of the information as possible. In this appendix, we describe the various techniques we use to make this conversion.

A.1. Rotcur Rotcur (Begeman 1987) is a standard algorithm to fit galaxy kinematics with a tilted-ring model. We used the implementation of rotcur in the NEMO package (Teuben 1995). Rotcur divides the galaxy into a set of narrow, concentric rings, and in each ring performs a nonlinear least squares fit to the function

vmodel(x, y)= 16 Simon et al.

−(x − x0) sin P A + (y − y0)cos P A v0 + vrot sin i , (A1) 2 2 2 (x − x0) + (y − y0) / cos i where v0 is the systemic velocity, vrot is the rotationp velocity, i is the inclination angle, P A is the angle between north and the receding side of the galaxy’s major axis, and (x0,y0) is the galaxy’s center. Each ring can thus contain up to six free 2 2 parameters (the central position requires two), and rotcur finds the best fit by minimizing i(vobs,i − vmodel,i) /wi , where wi is the weight ascribed to each point. We weight each point by the cosine of the angle between the point and the major axis, automatically deemphasizing points near the minor axis, so it is not necessary to discardP points within some angle of the minor axis. Rotcur’s most serious weakness is that it can only model rotational motions. To create the rotcur rotation curve, we used the best-fit center and systemic velocity that we determined with ringfit. Because the position angle must be a function of radius if the galaxy is modeled with purely rotational motions, we first ran rotcur with both the rotation velocities and the position angle free to vary to determine P A(r). We then used this description of the position angle as an input to rotcur, and ran it again with only the rotation velocities as free parameters. The rotation curve produced in this way is displayed in Figure 7b. We did not allow rotcur to fit for the inclination angle because it was apparent early on that the rotation curve of NGC 2976 is essentially solid-body, which means that dvrot/dr is small. Therefore, the inclination angle and the rotation velocities are degenerate in Equation A1, making the kinematic inclination angle poorly determined. We judged that the inclination angle was unlikely to differ significantly from the photometric value anyway, so the safest course was to leave the inclination fixed at 61.5◦.

A.2. Ringfit In addition to rotcur we constructed tilted-ring models with our own routine, ringfit. The purpose of this exercise was twofold: first, to compare the results from rotcur with those from a completely independent program and make sure that the answers agreed, and second, to fit for radial motions in the plane of the galaxy (inflow or outflow) instead of just assuming that the observed velocity field was due only to rotation. The ringfit fitting function is similar to Equation A1, except that we add an extra term to allow for radial velocities, and we do not fit for the PA, inclination, or the center. Thus, we can drop the explicit mention of the PA, x0, and y0, and write

vmodel = v0 + vrot sin i cos θ + vrad sin i sin θ, (A2) where cos θ is equal to the fractional expression that follows vrot sin i in the second term on the right hand side of Equation A1, and the free parameters in each ring are v0, vrot, and vrad. The solution is then determined with a linear least squares fit. The inclination, PA, and central position must be specified as inputs, but they can also be solved for by running ringfit with a grid of input parameters and minimizing χ2. We have verified that rotcur and ringfit give indistinguishable results when the same input parameters and weighting function are used.

A.3. Rotcurshape We also employed the NEMO routine rotcurshape, which is closely related to (and based on) rotcur. Rotcurshape dispenses with dividing the galaxy into rings and instead fits the whole velocity field at once. In addition to calculating the best-fit values for the PA, inclination, systemic velocity, and center, rotcurshape also assumes a functional form for the rotation curve/density profile (e.g., power law, NFW, pseudoisothermal, etc.) and solves for the free parameters of that function. One advantage of this approach is that near the center of the galaxy, where the velocities may be changing rapidly with radius, all of the data points are not artificially placed at the same radius (as was necessary with rotcur, where every point with r < 8′′ was in the same ring). Another is that the kinematic parameters of the galaxy and the parameters of the fitting function are determined in a single step. This makes it straightforward to measure the global agreement between the model and the data. For a power law rotation curve, the results from rotcurshape are nearly identical to the ones we derive by running rotcur or ringfit and then fitting a power law to the resulting rotation velocities.

B. nfw and power law density profiles Navarro et al. (1996) showed that CDM halos have density profiles of the form

ρ(r) δc = 2 , (B1) ρcrit (r/rs)(1 + r/rs) 2 −29 −3 where ρcrit = 3H /8πG ∼ 10 g cm is the critical density, δc is the halo overdensity, and rs is the scale radius (simulations suggest rs ∼ 2.5 kpc for a galaxy the size of NGC 2976). For r ≪ rs, Equation B1 clearly reduces to −1 ρ ∝ r . The commonly-discussed concentration parameter c is the ratio of the virial radius of the halo (r200, the radius enclosing a mean density of 200 times the background density) to the scale radius. In the simulations analyzed by NFW the concentration parameter varied from ∼ 7 for galaxy clusters up to ∼ 16 for large galaxies. Later studies at lower 10 masses found a median concentration of c = 20.5 for 3 × 10 M⊙ halos (Bullock et al. 2001). Other numerical simulations have resulted in slightly different profile shapes. For example, Moore et al. (1999b) argue that CDM halos exhibit steeper central cusps when simulated at higher resolution; their best-fitting functional form is similar to that of NFW, except that both terms in the denominator of the right hand side of Equation B1 are raised to the 1.5 power, resulting in a Dark Matter Halo of NGC 2976 17

ρ ∝ r−1.5 central density profile. Most subsequent studies in the literature have found central slopes that are bracketed by the NFW and Moore profiles (e.g., Jing & Suto 2000; Ghigna et al. 2000; Klypin et al. 2001; Power et al. 2003). It is noteworthy that no set of simulations has found central density profiles that are shallower than ρ ∝ r−1, although Taylor & Navarro (2001) presented analytical arguments for a ρ ∝ r−0.75 central slope. Since we are mostly interested in power law fits to the rotation curve, we also note that for a spherical mass distribution, −α a density profile ρ = ρ0(r/r0) implies that

− 4πGρ r2 r (2 α)/2 v = 0 0 , (B2) rot 3 − α r s  0  β and correspondingly, a rotation curve that can be fit by a power law vrot = v0(r/r0) yields a density profile

− (2β + 1)v2 r 2β 2 ρ = 0 . (B3) 4πGr2 r 0  0  A galaxy with a constant density halo thus has a linear (vrot ∝ r) rotation curve, while the rotation curve associated with −1 1/2 an NFW ρ ∝ r central density profile is vrot ∝ r .

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Table 1 NGC 2976 Surface Brightness Profiles

b Filter Integrated Central Surface Brightness Inner Disk Outer Disk µsky a −2 b ′′ c ′′ −2 Magnitude µ0 [mag arcsecond ] Scale Length [ ] Scale Length [ ] [mag arcsecond ] B 10.71 21.31 ± 0.01 79.3 ± 1.5 34.1 ± 0.4 22.10 V 10.14 20.69 ± 0.01 72.8 ± 0.8 33.6 ± 0.2 21.28 R 9.66 20.21 ± 0.01 71.2 ± 0.7 34.4 ± 0.2 20.71 I 9.19 19.73 ± 0.01 69.8 ± 0.7 33.1 ± 0.1 19.68 J 8.29 18.88 ± 0.03 71.8 ± 4.0 34.3 ± 2.9 16.01 H 7.71 18.24 ± 0.04 70.2 ± 5.5 31.1 ± 3.9 13.98 Ks 7.48 18.03 ± 0.06 69.6 ± 7.2 31.5 ± 5.0 13.47

aThese magnitudes are measured within an elliptical aperture with a semimajor axis of 172′′ on our Lowell and 2MASS images. The galaxy does extend to somewhat larger radii on the Keck images, so we have certainly underestimated the flux here. The Keck data suggest that the Lowell magnitudes should be made ∼ 4 % brighter, although if the galaxy is more extended even than those images reveal, the true correction could be slightly larger. bCentral surface brightnesses and inner scale lengths were calculated from the light distribution between 10′′ and 70′′. cOuter scale lengths were calculated from the light distribution outside 100′′ for B, V, and R (where there was a visible transition region between the inner and outer disks), and outside 70′′ for the near-infrared bands (where there was no transition region).

Note. — We have applied Galactic extinction corrections to these data. Internal extinction corrections have not been applied, but our adopted values for the internal extinction are given in the text (§2.5) if the reader wishes to use them. Dark Matter Halo of NGC 2976 19

Table 2 Rotation Curve Data

a b b,c b,c d e f g Radius vrot vrad vsys v∗,rot vHI,rot vCO,rot ∆vdrift [′′] [km s−1] [km s−1] [km s−1] [km s−1] [km s−1] [km s−1] [km s−1] 6.2 6.8 ± 0.4 ± 3.6 3.8 ± 0.2 ± 3.3 −0.1 ± 0.1 ± 3.1 20.6 ± 0.5 0.9 2.4 4.7 10.0 9.5 ± 0.3 ± 4.3 8.4 ± 0.1 ± 3.8 −2.1 ± 0.1 ± 4.0 23.8 ± 0.6 1.5 4.1 4.4 14.0 14.0 ± 0.2 ± 3.1 12.4 ± 0.1 ± 2.8 −3.8 ± 0.1 ± 2.7 27.9 ± 0.7 2.1 5.0 3.1 18.1 19.8 ± 0.1 ± 5.5 14.2 ± 0.1 ± 2.4 −2.8 ± 0.1 ± 1.6 33.4 ± 0.8 2.5 5.0 1.9 22.1 26.1 ± 0.1 ± 3.9 15.7 ± 0.1 ± 2.7 −1.1 ± 0.1 ± 1.1 38.1 ± 0.8 2.8 5.1 1.0 26.0 28.7 ± 0.1 ± 3.5 15.4 ± 0.1 ± 2.7 −1.0 ± 0.1 ± 1.0 41.2 ± 0.7 3.4 5.0 0.4 30.0 28.7 ± 0.1 ± 2.7 13.0 ± 0.1 ± 2.6 −1.3 ± 0.1 ± 1.2 46.8 ± 0.7 3.8 4.9 −0.1 34.0 31.7 ± 0.1 ± 2.5 10.6 ± 0.1 ± 2.4 −0.5 ± 0.1 ± 1.7 50.0 ± 0.6 4.2 4.8 −0.5 38.0 35.5 ± 0.1 ± 2.3 9.5 ± 0.1 ± 2.3 0.1 ± 0.1 ± 1.5 52.9 ± 0.5 5.0 4.7 −0.8 42.0 39.4 ± 0.1 ± 2.1 8.0 ± 0.1 ± 2.7 0.2 ± 0.1 ± 1.3 57.0 ± 0.4 5.9 5.1 −1.1 46.0 42.9 ± 0.1 ± 2.3 7.1 ± 0.1 ± 3.1 −0.1 ± 0.1 ± 1.3 60.5 ± 0.2 6.4 4.9 −1.3 50.0 46.0 ± 0.1 ± 2.6 6.7 ± 0.1 ± 3.4 −0.8 ± 0.1 ± 1.5 64.0 ± 0.1 7.2 4.6 −1.5 54.0 49.1 ± 0.1 ± 3.1 5.5 ± 0.1 ± 3.5 −1.8 ± 0.1 ± 1.9 65.6 ± 0.1 7.9 5.2 −1.5 58.1 49.2 ± 0.1 ± 3.2 4.6 ± 0.1 ± 3.6 −3.0 ± 0.1 ± 2.2 67.9 ± 0.3 8.6 5.0 −1.5 62.1 51.4 ± 0.1 ± 3.2 5.3 ± 0.1 ± 3.1 −3.4 ± 0.1 ± 1.8 71.9 ± 0.4 9.6 5.1 −1.3 66.0 57.2 ± 0.1 ± 3.1 6.0 ± 0.1 ± 3.0 −3.8 ± 0.1 ± 1.7 73.2 ± 0.6 11.1 5.9 −0.8 69.9 63.8 ± 0.1 ± 2.9 5.4 ± 0.1 ± 3.6 −3.0 ± 0.1 ± 1.5 76.3 ± 0.8 13.2 5.9 −0.3 73.9 67.8 ± 0.1 ± 2.8 3.7 ± 0.1 ± 4.7 −1.7 ± 0.1 ± 1.4 80.7 ± 1.0 15.0 6.0 0.4 77.9 69.8 ± 0.1 ± 3.3 2.1 ± 0.1 ± 5.8 0.0 ± 0.1 ± 1.7 83.5 ± 1.2 16.7 6.2 1.1 81.9 71.7 ± 0.1 ± 3.9 1.5 ± 0.1 ± 6.6 2.3 ± 0.1 ± 2.2 84.9 ± 1.4 18.5 6.4 1.9 85.9 74.1 ± 0.1 ± 4.4 0.0 ± 0.2 ± 7.9 3.5 ± 0.1 ± 2.7 84.5 ± 1.6 20.2 6.5 2.7 89.9 76.7 ± 0.1 ± 4.8 −3.8 ± 0.3 ± 9.1 2.7 ± 0.1 ± 2.8 83.5 ± 1.8 22.1 6.7 3.4 94.0 79.9 ± 0.2 ± 4.4 −8.3 ± 0.4 ± 12.4 0.1 ± 0.1 ± 2.8 85.4 ± 2.0 23.6 6.9 4.1 100.6 83.6 ± 0.6 ± 3.3 0 0 84.5 ± 2.3 25.7 7.1 5.0 111.1 83.9 ± 0.6 ± 3.9 0 0 81.2 ± 2.8 25.1 7.5 5.7 120.7 88.7 ± 0.6 ± 4.4 0 0 80.8 ± 3.3 25.3 7.8 4.6 131.2 85.3 ± 0.8 ± 5.3 0 0 77.6 ± 3.8 25.4 8.1 3.2

aTo convert to pc, multiply by 16.7. bFitted velocities are given as value ± statistical error ± systematic error. cRadial velocities and systemic velocities were fixed at zero for the outer four rings, where a lack of velocity field information away from the major axis limited our ability to constrain them. d Stellar velocities are given for the case of M∗/LK= 1.0M⊙/L⊙K. To get the stellar velocities for a different stellar mass-to-light ratio, multiply the tabulated values by M∗/LK. The listed uncertainties include only statistical errors. p eThe uncertainties on the H I rotation velocities are not known because we do not have access to the original data, but are probably about 10 - 20 %. f The uncertainties on the CO rotation velocities are quite high because the CO-H2 conversion factor is not known accurately. Since the CO rotation velocities are so small, this uncertainty is unimportant. gThis column gives the asymmetric drift correction to the rotation curve (see §6.2.7). To correct for asymmetric drift, add the values in this column to the observed rotation velocities in column 2. 20 Simon et al.

Table 3 Stellar Mass-to-Light Ratio Predictions

c Color Mean Inner Predicted M∗/LK Predicted M∗/LR Mean Outer a,b b,d Disk Color [M⊙/L⊙K][M⊙/L⊙R] Disk Color B − V 0.53 0.45 0.97 0.60 B − R 0.98 0.46 1.03 1.10 V − I 0.87 0.47 1.04 0.93 V − J 1.72 0.49 1.11 1.85 V − H 2.31 0.50 1.16 2.33 V − K 2.48 0.49 1.13 2.53

aCalculated for 10′′ ≤ r ≤ 70′′. bNote that these colors have been corrected for Galactic extinction and internal extinction. The Galactic extinction is taken from Schlegel et al. (1998) and the internal extinction corrections are given in the text. cWe use the predictions for the formation epoch with bursts of star formation model, assuming a scaled Salpeter initial mass function, as described in Bell & de Jong (2001). dCalculated for 100′′ ≤ r ≤ 172′′.

Table 4 Central Positions for NGC 2552

Method of α (J2000.0) δ (J2000.0) Distance From Reference Determining Center Nucleus [′′] nucleusa 08h19m20.4s 50◦00′33′′ 0.0 1 outer isophotesb 08h19m20.4s 50◦00′36′′ 2.7 2 outer isophotesc 08h19m19.7s 50◦00′32′′ 6.8 3 NED 08h19m20.1s 50◦00′25′′ 8.5 4 SIMBAD 08h19m19.6s 50◦00′28′′ 9.2 5

aThis position is also the center of the inner isophotes. bMeasured by Swaters (1999) from their photometry. cMeasured by Swaters & Balcells (2002) from their photometry.

References. — 1, B¨oker et al. (2003); 2, van den Bosch & Swaters (2001); 3, Swaters et al. (2002); 4, Falco et al. (1999); 5, Cotton et al. (1999). Dark Matter Halo of NGC 2976 21

Fig. 1.— (a) BVR composite image of NGC 2976 from the 1.8 m telescope at Lowell Observatory. Exposure times were 10 minutes in B and 5 minutes in V and R. Note the distinct lack of a bulge, a bar, or any spiral structure. (b) BVR composite image of NGC 2976 with integrated intensity CO contours overlaid. Note how well the CO traces out the optical dust lanes. The dashed circle shows the extent of the BIMA primary beam. The contour levels are 0.35, 0.70, 1.4, 2.1, and 2.8 Jy km s−1 inside the primary beam, and a single contour at 1.4 Jy km s−1 is shown outside the primary beam. For these observations, 0.35 Jy km s−1 corresponds to a molecular hydrogen column density 20 −2 ′′ ′′ of 2 × 10 cm (assuming that the Galactic CO-H2 conversion factor is valid in NGC 2976). The BIMA synthesized beam (5. 2 × 6. 0) is shown in the upper left corner. 22 Simon et al.

Fig. 2.— DensePak fiber layout. The four outlying fibers are the sky fibers. Since they are located only about 1′ from the main array, in some cases the sky spectra were contaminated by emission from the target galaxy. Dark Matter Halo of NGC 2976 23

Fig. 3.— Continuum-subtracted Hα image of NGC 2976. This 4′ × 4′ image consists of two 1200s exposures on the Lowell 1.8 m telescope that have been combined to cover the whole galaxy. The images were taken through a 32 A-wide˚ filter, and the continuum was removed by appropriately scaling and subtracting images taken through a narrow-band filter centered at 6441 A.˚ The black contours represent integrated CO intensity, as in Figure 1b. The white dashed rectangles overlaid on the image show the intended locations of our DensePak pointings (these have not been corrected for pointing errors; see Section 2.2 and Figure 4a), with one row or column of fibers overlapping between every pair of adjacent pointings. The artifacts at (α, δ)= (09h46m54s, 67◦56′35′′) and (α, δ)= (09h47m07s, 67◦54′26′′) are caused by masking out residuals from bright stars. 24 Simon et al.

V (km s−1) HEL 80

100 a) b) 60 80

60 40

40 20 20 0 0 Dec. Offset (") −20 −20

−40 −40 −60 −60 −80

−100 −80 50 0 −50 50 0 −50 R.A. Offset (") R.A. Offset (") Fig. 4.— (a) Hα velocity field from DensePak observations. The contours represent Hα intensity from the image displayed in Figure 3. (b) CO velocity field from BIMA observations. The contours represent integrated CO intensity, as in Figure 1b. The angular resolution of each dataset is shown in the upper left corners. Dark Matter Halo of NGC 2976 25

Galactocentric Radius (pc) 500 1000 1500 2000 2500

18

19 )

−2 K s 20

21 J

22

23

24

Surface Brightness (mag arcsec I R 25 V

26 B 0 20 40 60 80 100 120 140 160 Galactocentric Radius (") Fig. 5.— Optical and near-infrared surface brightness profiles of NGC 2976. For the J and Ks profiles we plot data points and error bars, but we omit the points for B, V, R, and I because they would obscure the error bars. The J and Ks data can be traced further out, but we do not plot the data beyond where the uncertainties reach a factor of 2 (0.75 mag). The H-band profile has also been left off for clarity; the error bars for H and Ks overlap at most radii. In each color, the nucleus, exponential inner disk, and exponential outer disk are all visible. In the optical filters, there is a transition region between the inner and outer disks where the colors are bluer than the disk values. Exponential fits to the I-band profile are shown by the solid black lines. The vertical dashed line at a radius of 70′′ emphasizes the breakpoint between the inner and outer disks. For central surface brightnesses and disk scale lengths, see Table 1.

Galactocentric Radius (pc) 0 500 1000 1500 2000 2500 3000 Stars (M/L=0.19) Atomic Gas 22 10 Molecular Gas ) −2

21 10 Hydrogen column density (cm

20 10 0 20 40 60 80 100 120 140 160 180 Galactocentric Radius (")

Fig. 6.— Surface densities of the stars and gas in NGC 2976. The H I and H2 surface densities do not include helium, so the stellar surface densities are divided by a factor of 1.3 to match. Of the baryonic components, the stars dominate the inner disk, but the H I is almost as important in the outer disk. The molecular gas surface density outside 40′′ is quite uncertain. 26 Simon et al.

Galactocentric Radius (pc) Galactocentric Radius (pc) 500 1000 1500 2000 500 1000 1500 2000 100 α H Rotation a) a) Combined Rotation b) b) 90 CO Rotation Combined Radial Hα Radial Combined Systemic Power−law fit 80 CO Radial Hα Systemic HI linewidth ROTCUR Rotation 70

60 ) −1 50

40

Velocity (km s 30

20

10

0

−10 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Galactocentric Radius (") Galactocentric Radius (") Fig. 7.— (a) Hα and CO velocity field decompositions from ringfit. The blue points represent rotation velocities, the red points represent radial velocities, and the green points are systemic velocities. Open symbols are from the Hα velocity field, and filled symbols are CO data. The rotational and radial velocities of the CO and Hα are consistent with each other. Because the number of independent CO data points is small, we reduced the number of degrees of freedom in the fit by fixing the systemic velocities. The error bars are only statistical errors, which substantially underestimate the true uncertainties. (b) Combined velocity field decompositions from ringfit. To create this rotation curve, we combined the Hα and CO data into a single velocity field. We then ran a Monte Carlo simulation in which the velocity field is fit many times, assuming a PA, inclination, and center position that are drawn randomly from the Gaussian distributions PA = −37◦ ± 5◦, i = 61.5◦ ± 3◦, and center = nucleus ± 2′′. The curves show the mean results from 1000 realizations of the simulation, and the shading that follows the curves represents 1σ systematic uncertainties in each of the plotted quantities. The thin black line is a power-law fit to the −0.27 rotation curve, corresponding to a density profile of ρTOT ∝ r . The cyan points are the rotcur rotation curve, showing the difference that arises if the radial velocities are not included in the fit. Note that although we have plotted the radial motions as positive velocities, whether they represent inflow or outflow cannot be determined without knowing which side of the galaxy is the near side. Dark Matter Halo of NGC 2976 27

50 r=4"−8" r=8"−12" r=12"−16"

0

−50 50 ) r=16"−20" r=20"−24" r=24"−28" −1

0

Velocity (km s −50 50 r=28"−32" r=32"−36" r=36"−40"

0

−50 −100 0 100 −100 0 100 −100 0 100 Position Angle θ (deg) Fig. 8.— Fits to the velocity field using ringfit. The observed velocities are plotted as a function of angle θ in the plane of the galaxy, where θ = 0 is the major axis. Data points with small error bars are from the Hα velocity field (and are all independent), and data points with large error bars are from the CO velocity field (and are not all independent; the error bars have been increased to account for this). The black curves show the rotational component of the fits (cos θ), and the red curves show the fits including both rotation and radial motions (sin θ). The displacement of the velocity maxima from θ = 0 illustrates the need for radial motions in the fits. At radii beyond 40′′, radial motions are not needed to obtain good fits to the data. 28 Simon et al.

V (km s−1) res 25

100 20 80 15 60 10 40 5 20 0 0

Dec. Offset (") −5 −20

−10 −40

−60 −15

−80 −20

−100 −25 50 0 −50 R.A. Offset (") Fig. 9.— Residual velocity field after subtracting ringfit model from the combined Hα and CO velocity fields. Hα data are shown by the circles, and the CO data are shown by the closely-packed square pixels. The rms of the residuals is 6.4 km s−1; 5.5 km s−1 if the small patch of probably spurious CO emission southwest of the galaxy at (R.A. offset, Dec. Offset) = (-45′′, -20′′) is excluded. Dark Matter Halo of NGC 2976 29

Galactocentric Radius (pc) Galactocentric Radius (pc) 500 1000 1500 2000 500 1000 1500 2000 90 a) b) 80

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V 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Galactocentric Radius (") Galactocentric Radius (") Fig. 10.— (a) Minimum disk rotation curve of NGC 2976. Here we assume that the dark matter is dynamically dominant over the baryons at all radii, so that the observed rotation velocities (black circles) are attributable entirely to the dark matter halo. This represents the cuspiest possible shape for the dark matter halo. The plotted error bars are combined statistical and systematic uncertainties. The rotation velocities due to H I and H2 are plotted as dashed and dotted curves, respectively. A power law fit to the rotation curve is shown by the solid black curve. The corresponding density profile is ρ ∝ r−0.27. Residuals from the fit are displayed in the lower panel, and 1σ and 2σ departures from the fit are represented by the shaded regions. (b) Maximum disk rotation curve of NGC 2976. In this case, we scale up the stellar disk (solid gray curve) as high as the observed rotation velocities (gray circles) allow. The stellar disk shown here has a mass-to-light ratio of 0.19 M⊙/L⊙K . This is the most massive stellar disk that can be present without making the dark matter density increase with radius, which is probably not physically realistic. After subtracting the rotation velocities due to the stars, the rotation velocities due to the H I (dashed curve), and the rotation velocities due to the H2 (dotted curve) in quadrature from the observed rotation curve, the dark matter rotation velocities are displayed as black circles. The two missing data points near the center of the galaxy had vrot < v∗, yielding ′′ ′′ imaginary vhalo. The solid black curve is a power law fit to the halo velocities (for 14

Fig. 11.— Dark matter density profile slope αDM as a function of the assumed K-band stellar mass-to-light ratio. The error bars represent the formal uncertainty in the value of αDM from the power law fit. The dashed gray lines show the upper and lower limits to the mass-to-light ratio that we consider reasonable based on the combination of the stellar population models and the kinematics. Note that for small values of M∗/LK the dark matter density profile is slightly steeper than the total density profile. This unusual effect is caused by the steep increase in the H I rotation curve at r > 60′′. 30 Simon et al.

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Fig. 12.— Point-by-point comparison of Hα and CO velocities. Although the line vHα = vCO provides a very good description of the data, with remarkably small scatter, there are still small systematic trends visible near the center of the galaxy and at positive velocities.

Fig. 13.— Density profile slopes derived from simulated longslit observations of NGC 2976. The filled circles indicate the derived value of the density profile slope αTOT for each offset from the major axis. The dashed line shows the value of αTOT from our analysis of the full velocity field (without radial motions, since they cannot be accounted for in longslit observations). The open circles represent the slopes that would have been derived had the correct (closest to the actual center) folding point been selected for each of the slits. For some slits, the value of αTOT is quite sensitive to the choice of the folding point. Note that the open symbol at (10,0.0) should actually be located at αTOT = −0.21, except that we do not allow negative values of αTOT because they are unphysical.